On the Beilinson fiber square
Benjamin Antieau Akhil Mathew Matthew Morrow Thomas Nikolaus
math.KT math.AG 14F30, 14F40, 19D55, 19E15




B. Antieau]Benjamin Antieau
Department of Mathematics, Northwestern University
antieau@northwestern.edu

A. Mathew]Akhil Mathew
Department of Mathematics, University of Chicago
amathew@math.uchicago.edu

M. Morrow]Matthew Morrow
CNRS, Institut de Mathématiques de Jussieu–Paris Rive Gauche, Sorbonne Université
matthew.morrow@imj-prg.fr

T. Nikolaus]Thomas Nikolaus
FB Mathematik und Informatik, Universität Münster 
nikolaus@uni-muenster.de



On the Beilinson fiber square
    [
    December 30, 2023
=============================




Using topological cyclic homology, we give a  refinement of Beilinson's
p-adic Goodwillie isomorphism between relative continuous -theory and cyclic homology. 
As a result, we generalize results of Bloch–Esnault–Kerz and Beilinson
on the p-adic deformations of -theory classes. 
Furthermore, we prove structural results for the Bhatt–Morrow–Scholze filtration on
TC and identify the graded pieces with the syntomic cohomology of Fontaine–Messing.







§ INTRODUCTION




 §.§ Fiber squares

For any ring R, one has its connective algebraic -theory (R) and its negative cyclic homology ^-(R); they are related via
the Goodwillie–Jones trace map  tr_GJ(R) →^-(R), often
interpreted and referred to as a Chern character, <cit.>.
Moreover, when R is a ℚ-algebra, the 
map tr_GJ induces an isomorphism on relative theories for
nilimmersions, via the following theorem of Goodwillie.

 
If I ⊆ R is a nilpotent ideal in an associative ℚ-algebra R, then the
commutative square

    (R)[r][d]^_GJ   (R/I)[d]^_GJ
    ^-(R)[r]   ^-(R/I)

is cartesian, i.e., the Goodwillie–Jones trace map induces an equivalence
_GJ(R,I)^-(R,I) on relative theories.
 


Here for a pair (R, I) with I ⊆ R an ideal, we write (R, I)
for the
fiber of (R) →(R/I), and similarly for other functors such as ^- and so on.

In order to extend Goodwillie's theorem to more general rings, 
one uses topological cyclic homology TC(R),
introduced in <cit.> in the p-complete case and in <cit.> integrally,
and the cyclotomic trace tr(R) →(R), which
refines the Goodwillie–Jones trace map.


    If I ⊆ R is a nilpotent ideal in an associative ring R,
    then the commutative square
    
    (R)[r][d]^   (R/I)[d]^
    (R)[r]   (R/I)
 is cartesian, i.e., the cyclotomic trace induces an equivalence
    tr(R,I) (R,I) on relative theories.
 

Topological cyclic homology is thus the primary tool in  
calculations of relative -theory (see for example <cit.>), but it is a significantly 
more complicated invariant than cyclic homology. 
However, recently Beilinson <cit.>
gave a version of Goodwillie's original result 
in a p-adic setting, when the ideal in question is (p) and R is assumed
to be complete along (p). 
The first goal of this paper is to construct a variant of the Chern character
and prove a  strengthening of Beilinson's theorem. 

Throughout this paper, we fix a prime number p. 
We use the convention that 
the modifier “ℤ_p” refers to p-adic completion of an object, and
“ℚ_p” to
the rationalization of the p-completion; for example (R; ℤ_p)
denotes the p-complete -theory of R, and (R; ℚ_p) denotes the
rationalization of (R;_p). 
Similarly, the modifier “ℚ” refers to rationalization. 
We denote by  
(resp. )  periodic cyclic (resp. cyclic) homology. 

[A]
For an associative ring R, there is a natural  
p-adic Chern character map 

    tr_crys(R/p; ℚ_p) →(R; ℚ_p)
 
which fits into a natural commutative square

    (R;_p)[r][d]^_GJ   (R/p;_p)[d]^_crys
    ^-(R;_p)[r]   (R;_p).

If R is commutative and henselian along (p) then this square is cartesian, thereby giving an equivalence (R, (p); ℚ_p) ≃Σ(R; ℚ_p). 
 

In <cit.>, 
Beilinson constructs a natural equivalence[Since the methods are different, we do not know if
our identification on fiber terms
 is the same 
as Beilinson's.]
^cts(R, (p); ℚ_p) ≃Σ(R; ℚ_p)
under the assumption that R is p-complete with bounded
p-power torsion,   R/p
has finite stable range,[The stable range of a ring R was defined
    in <cit.> (see also <cit.>) and is sometimes, as
    in <cit.>, called the stable rank.]
    and the relative -theory term (R, (p)) 
is replaced by the “continuous” relative -theory
^cts(R, (p)) = (R/p^n,
(p)); this replacement does not affect the conclusion if R is 
commutative 
thanks to <cit.>. 
Beilinson's arguments  rely on some p-adic Lie theory. 


In this
paper, we will construct the map 
(<ref>) 
using the description of topological
cyclic homology from Nikolaus–Scholze <cit.>, as a consequence of
Bökstedt's calculation of (𝔽_p). 
Together with  the rigidity results of Clausen–Mathew–Morrow <cit.>, 
we explain a short, homotopy-theoretic proof of Theorem [thm:a]A. In
fact, Theorem [thm:a]A and all the corollaries listed below hold for
any (possibly non-commutative) ring R if we replace -theory by  (see
<Ref> and <Ref>); the henselian condition is only needed to
translate between -theory and .

Next, we observe some consequences of and complements to Theorem [thm:a]A.
In <cit.>,  slightly more than an equivalence of
rational spectra (R,(p);_p)Σ(R;_p) is proved: there is a natural zig-zag of 
“quasi-isogenies” of spectra before inverting p.    By definition, a
quasi-isogeny is a map which is an equivalence up to uniformly bounded denominators in any finite range of degrees.
We also obtain the same conclusion in 
our setting and can keep track of the denominators at least in some
range.

[B] 

Let R be a commutative ring which is henselian along (p). Then there is a natural
zig-zag of quasi-isogenies between (R, (p); ℤ_p) and Σ(R, (p);
ℤ_p). If R is moreover p-torsion free, then there are
isomorphisms π_i(R,(p);_p)≃π_iΣ(R,(p);_p) for i≤ 2p-5.
 

A similar result for an arbitrary nilpotent ideal, albeit in a smaller range of
degrees (depending on the exponent of nilpotence of the ideal), is proved in <cit.>. The argument we use here seems to be special to the ideal
(p). 

As explained above, one could formulate Theorem [thm:a]A entirely in
the language of topological cyclic homology, completely avoiding the mention
of -theory. At some point in the proof, however, we translate back into
-theory and use a homology argument. Therefore,  we also offer an alternative purely cyclotomic proof of this step. 
This relies on a study of 
quasi-isogenies in the homotopy theory of cyclotomic spectra, based on the
t-structure introduced by Antieau–Nikolaus <cit.>.  
The key step is an extension of a theorem of Geisser–Hesselholt <cit.>
and Land–Tamme <cit.>. 

[C]
    Let f  A→ A' be a map of connective associative ring
    spectra. Suppose that
    
        
  (1) f is a quasi-isogeny of spectra and
        
  (2) the map π_0(f)π_0(A)→π_0(A')
            is surjective with nilpotent kernel.
    
    Then (A; ℤ_p)→(A'; ℤ_p) is a quasi-isogeny in cyclotomic spectra
    and in particular the induced map (A;_p)→(A';_p) is a
    quasi-isogeny.




 §.§ p-adic deformations of -theory classes

In our first main application of  Theorem [thm:a]A, we generalize work of
Bloch–Esnault–Kerz <cit.> and Beilinson <cit.> on the formal
p-adic deformation of
rational -theory classes. 
Let us first recall the motivation for their work. 

Fix a complete discretely valued field K of mixed characteristic (0, p) with ring
of integers 𝒪_K and perfect residue field k as well as a proper smooth
scheme X →(𝒪_K) with special fiber X_k and generic fiber
X_K. 
Given X, we can consider the algebraic
de Rham cohomology H^*_(X_K/K) of the generic fiber, together with its
Hodge filtration Fil^≥⋆ H^*_(X_K/K); these are
finite-dimensional K-vector spaces, and arise as the 
cohomology groups of objects Fil^≥⋆ R Γ_(X_K/K)
in the derived category of K. 


As usual, we have a Chern character 

    ch_0(X; ℚ) 
    ↠_0(X_K; ℚ) → 
    H^even_dR(X_K/K).

A foundational motivating question is to determine the image of this map: in
other words, to determine which cohomology classes come from algebraic cycles
on X_K (or equivalently on X). 
A conjecture of Fontaine–Messing, a p-adic analog of the variational Hodge
conjecture of Grothendieck <cit.>,  predicts 
that this question can essentially be reduced from mixed to equal characteristic. 

To formulate the conjecture, we consider also the 
(absolute)
crystalline cohomology 
H^*_(X_k) of the special fiber, a family of finitely generated
W(k)-modules. 
By the de Rham-to-crystalline
comparison <cit.>, we have an isomorphism  
H^*_(X_k) ⊗_W(k) K  H^*_(X_K/K) .
Finally, we have the crystalline Chern character map <cit.>,

    ch_crys_0(X_k) →⊕_i ≥ 0
    H^2i_(X_k; ℚ_p),

leading to a commutative diagram

    _0(X; ℚ)  [d]^ch[r]    _0(X_k; ℚ)
    [d]^ch_crys
    
    H^even_dR(X_K/K) [r]^<<<<≃[r]      
    H^even_crys(X_k) ⊗_W(k) K.
 


  Let 
α∈ H^even_dR(X_K/K). Then α belongs to
the image of the Chern character from _0(X; ℚ) if and only if


  * the image of α under the 
de Rham-to-crystalline isomorphism in 
H^even_crys(X_k) ⊗_W(k) K belongs to the image
of the crystalline Chern character from 
_0(X_k; ℚ) and

  * the class α belongs to ⊕_i Fil^≥ i
H^2i_dR(X_K/K) ⊆ H^even_dR (X_K/K). 


 
For further details and arithmetic applications of the p-adic variational Hodge
 conjecture, we refer
to <cit.>.


Motivated by Conjecture <ref>,
 Bloch–Esnault–Kerz <cit.> considered  
the following p-adic deformation question, which starts with a _0-class
on the special fiber (rather than a cohomology class) and asks when it lifts
infinitesimally. 

[The p-adic deformation problem] 
Given the data as above, define 
the “continuous” -theory 
    ^cts(X)
    def=(X/π^n)
    ,
 
where π is a uniformizer of _K.
Given a class x ∈_0(X_k; ℚ), when does it belong to the image of 
the reduction map from 
the continuous -theory
_0^cts(X; ℚ) = π_0
(^cts(X))_ℚ? 
 



Since the map (X) →^cts(X) is generally not an equivalence,
the p-adic deformation problem does not imply <Ref>. 
However, the p-adic deformation problem 
is a (pro)-infinitesimal one, so it can be
studied using methods of topological cyclic homology. 
Using the Beilinson fiber square, we answer the p-adic deformation problem as
follows; 
in <cit.>, this result is proved for  a smooth projective scheme of
dimension d < p + 6 when K is unramified. 

[D]  
Let X be a proper smooth scheme over 𝒪_K. 
A class x ∈_0(X_k; ℚ) lifts to _0^cts(X;
ℚ) if and only if 
ch_crys(x) ∈⊕_i ≥
0H^2i_crys(X_k; ℚ_p) is carried by the de
Rham-to-crystalline comparison isomorphism to a class in ⊕_i ≥ 0Fil^≥ iH^2i_(X_K/K) ⊆⊕_i ≥ 0
H^2i_(X_K/K).
 

Our main observation is that 
Theorem [thm:a]A together with
Hochschild–Kostant–Rosenberg comparisons between cyclic and
de Rham cohomology yield a fiber square

    ^cts(X; ℚ) [d] [r]    (X_k; ℚ) [d]
    
    ∏_i ∈ℤFil^≥ i R Γ_dR(X_K/K)[2i] [r]    ∏_i ∈ℤ  R Γ_dR(X_K/K)[2i].

Moreover, on _0, one checks that the vertical map on the right-hand side induces 
 the crystalline Chern character (<ref>), at least up to
 scalars, implying the result. 
For this argument, it is crucial that one has the fiber square 
(<ref>), rather than a fiber sequence alone. 

One can also generalize the above questions to higher -theory. 
In  <cit.>, 
the Beilinson fiber sequence is used to prove that if x ∈_j(X_k;
ℚ), then 
there exists a natural obstruction class 
in 
⊕_i ≥ 0 H^2i-j_dR(X_K)/Fil^≥ i
H^2i-j_dR(X_K)  which vanishes if and only if x lifts to the
continuous -theory ^cts_i( X; ℚ);
however, <cit.>
does not 
identify the class
with the crystalline Chern character for i  = 0. 
Here we also extend this result to an
arbitrary quasi-compact and quasi-separated (qcqs) scheme with bounded p-power
torsion, using p-adic derived de Rham
cohomology <cit.> and results of <cit.>.

[E]
    Let X be a qcqs scheme with
	 bounded p-power torsion. For each
     n we write X_n for X×_/p^n, and put ^cts(X)
def=(X_n).
	 Given a
    class x∈_j(X_1;), there is a natural class
    
    c(x)∈⊕_i≥
        0H^2i-j(ŁΩ_X/ŁΩ^≥ i_X)_ℚ_p,
 where 
	ŁΩ_X is the p-adic derived de Rham cohomology of X with the derived Hodge
	filtration ŁΩ^≥⋆_X. The class
    x lifts to _i^cts(X; ℚ) if and only if
    c(x)=0.







 §.§ The motivic filtration on TC

In 
<cit.>, 
 Bhatt–Morrow–Scholze 
discovered a fundamental additional structure on the p-adic topological cyclic
homology TC(-; ℤ_p) of p-adic commutative rings: a
“motivic filtration”
Fil^≥⋆TC( -; ℤ_p)
on TC(-; ℤ_p)
with associated graded terms 
denoted
 ℤ_p(i)[2i]. 
 The objects ℤ_p(i) thus obtained are related to integral p-adic Hodge
 theory and can be defined (independently of topological cyclic homology) as a type of filtered Frobenius invariants on the
 prismatic cohomology <cit.>. 
 They are known explicitly in some cases: in characteristic p> 0 they can be identified
 with logarithmic de Rham–Witt sheaves (up to a shift), and for formally smooth
 algebras over 𝒪_C (for C a complete algebraically closed
 nonarchimedean field), they can be identified with truncated p-adic nearby
 cycles. 


Recall also that for p-adic (commutative) rings, TC(-;
ℤ_p) is p-adic étale -theory in
nonnegative degrees <cit.>. Therefore, it is expected (but not known
in mixed characteristic) that 
the filtration 
Fil^≥⋆TC( -; ℤ_p)
is the étale sheafified motivic filtration on algebraic -theory, and that
the ℤ_p(i) are p-adic étale motivic cohomology, at least where all of
these objects are defined. 
One also has constructions of Schneider and Sato <cit.> of “p-adic
étale Tate twists,” which satisfy a type of arithmetic duality. 
In general, one expects that the ℤ_p(i) should be related to
important foundational questions in arithmetic geometry and -theory. 
An advantage of the construction of 
Fil^≥⋆TC( -; ℤ_p)
and the ℤ_p(i) as in <cit.>
is that it
works in a much more general setting (for the quasisyntomic rings; see
Section <ref>
below for a review) than existing approaches to motivic
cohomology. Moreover, its definition is extremely direct: it is simply a sheafified
Postnikov tower (albeit for a “large” topology). 
 
Using Theorem [thm:a]A, 
we will give a description of the ℤ_p(i) for i ≤ p-2 and of
the ℚ_p(i) for all i in terms of syntomic
cohomology as considered by Fontaine–Messing <cit.> and Kato <cit.>.
In particular, this construction gives a description of the ℤ_p(i)
(with the above restrictions) that relies only on derived de Rham theory, rather
than prismatic theory. 
Our result is an analog of 
a result of Geisser <cit.> for étale motivic cohomology for
smooth schemes over Dedekind rings. 

To formulate the result, we write L Ω_R for the p-adic derived de Rham
cohomology for a commutative ring R equipped with its derived Hodge filtration
L Ω_R^≥⋆ <cit.>. 
The object L Ω_R carries a crystalline Frobenius φ
L Ω_R → L Ω_R. 
For i < p, 
one has a “divided” Frobenius φ/p^i L Ω_R^≥ i→ L
Ω_R. Using the techniques of <cit.> (in particular, quasisyntomic
sheafification) applied to the Beilinson fiber square, we deduce our next
theorem.

[F]
Let R be a quasisyntomic ring.

    
  (1)  
For each i ≥ 0, there is an identification

    ℚ_p(i)(R) ≃fib( φ - p^i L Ω_R^≥ i→ L
    Ω_R)_ℚ_p.


  (2)
For i ≤ p-2, there is an identification 

    ℤ_p(i)(R) ≃fib( ϕ/p^i - id L
    Ω_R^≥ i→ L Ω_R ).


 

We explicitly analyze Theorem [thm:f]F in
three cases in which one has alternate descriptions of the ℤ_p(i): rings of integers in p-adic fields, perfectoid rings, and
formally smooth 𝒪_C-algebras where C is an algebraically closed,
complete nonarchimedean field of mixed characteristic.
The first case recovers classical calculations of the rational p-adic -theory of p-adic fields;
the second case recovers the fundamental exact sequence in p-adic
Hodge theory; the last case 
recovers results of Colmez–Nizioł 
<cit.> on p-adic vanishing cycles, albeit only in the formally smooth case.


Finally, Theorem [thm:f]F  provides a complete computation of low-degree or rationalized
TC in terms of syntomic cohomology. This computation relies on the  following 
 connectivity estimate about the
ℤ_p(i) and about the filtration on TC(-;
ℤ_p).  
The estimate for algebras over a perfectoid ring is stated in
<cit.>; the argument for all quasisyntomic rings relies on the use of
relative topological Hochschild homology and the spectral sequence of
Krause–Nikolaus <cit.>. 

[G]
If R is a quasisyntomic ring, then 
ℤ_p(i)(R) ∈ D^≤ i+1( ℤ_p). 
If R is
w-strictly local (e.g., strictly henselian local), then 
ℤ_p(i)(R) ∈ D^≤ i(
ℤ_p). 
Consequently, Fil^≥ iTC(R; ℤ_p) is
concentrated in homological degrees ≥ i-1 (and pro-étale locally ≥
i). 
 

[H]
If R is any commutative ring, then there is a natural equivalence

    (R; ℚ_p) ≃⊕_i ≥ 0fib( φ -
    p^i L Ω_R^≥ i→ L Ω_R)_ℚ_p.

 




 §.§ Notation
 
Throughout, we write  for the ∞-category of spectra. 
Given a ring R, we write D(R)
for the derived ∞-category of R.


We will use homological indexing conventions indicated with a subscript
when referring to spectra and
cohomological indexing conventions indicated with a superscript when referring to objects of the derived
category. 
For instance, given n, we
write _≥ n (resp. _≤ n) for spectra with homotopy groups
concentrated in degrees ≥ n (resp. ≤ n); 
we write D(R)^≤ n (resp. D(R)^≥ n) for objects of D(R) with cohomology groups concentrated in
degrees ≤ n (resp. ≥ n).  



We write (R) for the Hochschild homology of R, always relative to
ℤ and always computed in a derived sense (also known as Shukla
homology), and we let (R) denote the topological Hochschild homology of
R. We write
^-(R)  = (R)^hS^1 for negative cyclic homology and (R) =
(R)^tS^1 for periodic cyclic homology. 
For a scheme X, we let LΩ_X denote its p-completed derived
de Rham cohomology (relative to ℤ) and 
ŁΩ_X^≥ i for the ith stage of the
(derived) Hodge filtration. 
We denote the Hodge-completed variants by ŁΩ_X
and ŁΩ^≥ i_X, respectively.



 §.§ Acknowledgments

We are very grateful to Peter Scholze for suggesting this question to us and for
sharing many insights.  We would also like to thank Johannes Anschütz,
Alexander Beilinson, Bhargav Bhatt, Hélène Esnault,   Lars Hesselholt,
Luc Illusie, Moritz Kerz,
Arthur-César Le Bras,
Wiesława Nizioł, and
Sam Raskin for helpful conversations. 
The third author would like to thank Uwe Jannsen, Moritz Kerz, and Guido Kings
for organising and inviting him to the 2014 Kleinwalsertal workshop on
Beilinson's paper <cit.>. 

Two referees provided detailed, invaluable comments on the paper which has been
improved as a result.

The first two authors would like to thank the University of Bonn and the University
of Münster for their hospitality. This material is based upon work supported by the National Science Foundation
under Grant No. DMS-1440140 while the first three authors were in residence at the
Mathematical Sciences Research Institute in Berkeley, California, during the
Spring 2019 semester.
The first author was supported by NSF Grant DMS-1552766.
This work was done while the second author was a Clay Research Fellow. The
fourth author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy 
EXC 2044 Ð390685587, Mathematics Münster: Dynamics–Geometry–Structure.





§ THE BEILINSON FIBER SQUARE




 §.§ Background


We review some background on the theory of  cyclotomic spectra and topological cyclic homology as in
<cit.>, of which we will  use the p-typical variant. This theory uses the ∞-category Fun(BS^1,) of spectra
equipped with S^1-actions. Given a spectrum X equipped with an
S^1-action, we can form the homotopy S^1-orbits X_hS^1, the homotopy
S^1-fixed points X^hS^1, and the S^1-Tate construction X^tS^1.
These are related by a natural fiber sequence Σ X_hS^1→
X^hS^1→ X^tS^1, which we will use constantly and without further
comment. See for example <cit.>.

 
We let  denote the symmetric monoidal, stable ∞-category of
cyclotomic spectra.[Our conventions are slightly different from
those of <cit.>, which requires a Frobenius map for each prime number
q and not only the fixed prime p. What we call a cyclotomic spectrum is called a
p-typical cyclotomic spectrum by <cit.>, and what we write as
 is written as _p in <cit.>. For p-complete objects
(the primary case of interest), there is no difference between a
cyclotomic spectrum in the sense of <cit.> and the definition used here.]
An object of  consists of a spectrum X
equipped with an S^1-action and an S^1-equivariant
map φ_p X →
X^tC_p called the cyclotomic Frobenius.

Given X ∈, we write
    
    ^-(X) = X^hS^1  and  (X) = X^tS^1.

We will define only p-complete  for X ∈  and will assume that X is bounded below.
We have two maps
^-(X;_p) →(X;_p) and φ^-(X;_p) →(X;_p).
By definition,  is the canonical map from S^1-invariants to the Tate
construction, and φ is induced from the Frobenius φ_p. 
The p-complete topological cyclic homology (X;_p), for X ∈ bounded below, can be
computed as the
fiber  of the difference of the two maps, i.e., 

    (X;_p) = fib(  - φ^-(X;_p) →(X;_p)).

 



    We will use throughout the basic fact that if X ∈ has underlying
    n-connective spectrum, then (X;_p) is (n-1)-connective (see for
    example <cit.>).



 


  * Given a ring R, we can form the topological Hochschild
homology (R) as a cyclotomic spectrum.   

  * Given a spectrum Y, 
we let Y^ be the cyclotomic spectrum where we view Y as a spectrum
with trivial S^1-action and with cyclotomic Frobenius
given by the natural map
Y → Y^hC_p→ Y^tC_p. 

  * For a spectrum X with S^1-action we get a cyclotomic spectrum by
    letting φ_p X → X^tC_p be zero (as an S^1-equivariant map). 

 

 
Let Y be a bounded below spectrum of finite type, meaning that each π_iY
is a finitely generated abelian group. If X ∈ is
p-complete and bounded below, then there is a natural equivalence
    
    (X ⊗_ Y^; ℤ_p) ≃(X;
    ℤ_p) ⊗_ Y.
 This is a
consequence of the fact that the functor (-; ℤ_p) commutes with 
geometric realizations of
connective cyclotomic spectra, since it is exact and carries n-connective
objects into (n-1)-connective objects. 
This even holds if Y is only assumed to be bounded below, as long as the
right side of (<ref>) is p-adically completed, by <cit.>.
 

As a simple exercise with the above definitions, we prove the following result
for use below. This has been used in other references as well, e.g.,
<cit.>. 



If X ∈ is bounded below and (X;_p) =0, then we have natural equivalences
(X;_p) ≃ 
(Σ X_hS^1)^∧_p ≃^-(X;_p). 

 
    In fact, the formula (<ref>) shows that (X;_p) = ^-(X;_p). 
    Now (X;_p) is the cofiber of the norm map (Σ X_hS^1)^∧_p →^-(X;_p).
    Since we have assumed (X;_p) =0, we have (Σ X_hS^1)^∧_p≃^-(X;_p).
    Combining the two identifications, we conclude. 
 

Next, we apply this to a specific crucial example.

[The cyclotomic spectrum ℤ_hC_p]
Recall first that the cyclotomic
trace (or a direct construction) gives a map  
    ℤ^→( 𝔽_p)
 
in , and that 
as objects of (BS^1, ) we have 
THH( 𝔽_p) ≃τ_≥ 0(
ℤ^tC_p) by <cit.>; here we obtain the S^1-action
on ℤ^tC_p via the sequence C_p → S^1  S^1. This is a refinement of (and deduced from) Bökstedt's calculation of (
𝔽_p). 
Consequently, 
there is a cofiber sequence in ,

    _hC_p→^→(_p),
 where _hC_p is
a cyclotomic spectrum with 
underlying spectrum with S^1-action ℤ_hC_p∈(BS^1, ). 

Note that 
(_hC_p)^tC_p 0 by the Tate orbit lemma <cit.>.
In particular, the map 
ℤ^triv→THH( 𝔽_p) induces an
equivalence on (-; ℤ_p), i.e., 
ℤ_p^tS^1≃THH(𝔽_p)^tS^1=(_p).
We obtain 
by <Ref> that TC( ℤ_hC_p; ℤ_p) ≃Σ (ℤ_p)_hS^1. 
Our next observation is that this remains true after tensoring with any bounded
below cyclotomic spectrum.


 

If X ∈(BS^1, ), then
(X ⊗_ℤ_hC_p)^tS^1
is p-adically zero (here we use the diagonal S^1-action).
 


    The spectrum (X ⊗_ℤ_hC_p)^tS^1 is a module over
    (ℤ^hC_p)^tS^1. Since (ℤ^hC_p)^tS^1 
    vanishes p-adically by the Tate fixed point lemma <cit.>, the lemma follows.
	 Alternatively, one can easily verify that (𝔽_p)_hC_p∈(BS^1, ), is induced from the
	 trivial subgroup, which forces the p-adic Tate vanishing. 
 

Combining Proposition <ref> and Lemma <ref>, we conclude that if X is any bounded below
cyclotomic spectrum, then there are equivalences

    (X ⊗_ℤ_hC_p; ℤ_p) ≃Σ((X
        ⊗_ℤ_hC_p)_hS^1)^∧_p ≃^-(X ⊗_ℤ_hC_p;_p).
 





 §.§ Pullback squares
 Next, we establish some 
pullback squares involving cyclotomic spectra and give a proof of
Theorem [thm:a]A. 



Let X ∈ be a bounded below cyclotomic spectrum. Then 
the commutative square

    ( X ⊗_ℤ^; ℤ_p) [d]  [r]    (X
        ⊗_(
    𝔽_p); ℤ_p)
    [d]  
    ^-(X  ⊗_ℤ^;_p) [r]    ^-(X ⊗_(𝔽_p);_p)

is cartesian,
where the horizontal maps arise from the map ℤ^→(𝔽_p) in  of Construction <ref> and the vertical maps are the
canonical maps (-;_p) →^-(-;_p) arising from the definition of
(-;_p).

Moreover, there is a natural fiber sequence 

    (Σ (X ⊗_ℤ_hC_p)_hS^1)^∧_p →(X
        ⊗_ℤ^; ℤ_p)  →(X ⊗_(𝔽_p);
    ℤ_p).

 

Since TC(Z; ℤ_p)  for a bounded below cyclotomic spectrum Z is an equalizer of two maps 
^-(Z;_p) ⇉(Z;_p), the statement that 
(<ref>) is cartesian follows from 
the fact that X ⊗_ℤ^→ X ⊗_(𝔽_p)
induces an equivalence on (-;_p), via 
<Ref>. 
Moreover, the fiber sequence (<ref>) then follows from 
(<ref>) via taking fibers, and using <Ref> again
to replace homotopy fixed points by homotopy orbits. 
Alternatively, to prove that (<ref>) is cartesian, one observes that the
fibers of the horizontal arrows
are (X ⊗_ℤ_hC_p; ℤ_p) and ^-(X
    ⊗_ℤ_hC_p;_p)
and these are naturally equivalent as in (<ref>). 



For every connective ring spectrum R we have a natural fiber sequence
of p-complete spectra

    Σ((R;_p) ⊗_ℤ_hC_p)_hS^1→(R;_p)⊗_→(R⊗__p;_p)



We apply  <Ref> to X = (R; ℤ_p). We have that (R)
⊗_(_p) ≃(R ⊗__p) which gives the
identification of the third term. For the identification of the term in the
middle we observe that   (X ⊗_ℤ^; ℤ_p) ≃(X; ℤ_p) ⊗_ℤ by 
(<ref>). 
Note finally that for bounded below spectra, tensoring with ℤ
preserves p-completeness as ℤ is finite type. 


Next, we study what happens in (<ref>) after rationalization. 

 

Let X ∈ be a bounded below, p-complete cyclotomic spectrum. 
Then there exists a natural map 
(X ⊗_(𝔽_p); ℤ_p) → (X ⊗_ℤ^)^tS^1 which fits into a natural commutative square
    
    (X ⊗_ℤ^; ℤ_p) [d] [r]    (X
            ⊗_(𝔽_p);
    ℤ_p) [d] 
    
    (X ⊗_ℤ^)^hS^1[r]     (X ⊗_ℤ^)^tS^1.

Moreover, this square 
becomes cartesian after inverting p. 
 
 
We can vertically extend the cartesian square (<ref>) 
via the canonical maps (-)^hS^1→ (-)^tS^1. 
In this case, as we saw earlier, the map 
(X ⊗_ℤ^)^tS^1→ (X ⊗_(𝔽_p))^tS^1 is an equivalence. 
Using this identification, we obtain the commutative square (<ref>). 
The fact that (<ref>) is cartesian after inverting p
follows from the facts that (<ref>) is cartesian and that
(X ⊗_(𝔽_p))^hS^1→ 
(X ⊗_(𝔽_p))^tS^1≃ 
(X ⊗_ℤ^)^tS^1 becomes an equivalence after inverting
p. 
 


 

For future reference, it will be helpful to give a more effective version of 
<Ref>. 
Consider the total cofiber (cofiber of horizontal cofibers) 
of the square (<ref>). This is given by Σ^2 ( X ⊗_(
𝔽_p))_hS^1 because (<ref>) is homotopy cartesian.
If X is connective, then it follows that the τ_≤ 2i of the total cofiber is 
annihilated by p^i, since τ_≤ 2i-2(𝔽_p) is
S^1-equivariantly annihilated by p^i: indeed, this follows by the
(S^1-equivariant) Postnikov
filtration, since each nonzero homotopy group of (𝔽_p) is in even
degree and is an
𝔽_p-vector space. 
 

Consequently, we can deduce the following fiber square, which is the basic
-theoretic result from which the Beilinson fiber square is a consequence. 
 

Let R be a ring (or more generally, a connective associative
Hℤ-algebra spectrum). 
Then there is a natural commutative square of spectra
    
    (R; ℤ_p) [d]  [r]    (R ⊗_𝕊𝔽_p)
    [d]  
    ^-(R; ℤ_p) [r]    (R; ℤ_p),

which becomes cartesian after inverting p. 
Aside from the right vertical arrow, all the maps are the canonical
ones. 
 
 
    Via (<ref>) for X = (R;_p), 
    we obtain a natural commutative diagram
    
    (R;_p)[d]   
    ((R) ⊗_ℤ^; ℤ_p) [d] [r]    ( R ⊗_𝕊𝔽_p)
        [d]  
    
        ((R; ℤ_p) ⊗_ℤ^)^hS^1[r] [d]     ((R;
        ℤ_p)
        ⊗_ℤ)^tS^1[d]
    ^-(R;_p)[r]   (R;_p),

    where we use the natural cyclotomic map
    (R;_p)→(R;_p)⊗_^
    and the natural S^1-equivariant map
    (R;_p)⊗_→(R;_p). The upper square is
    cartesian after inverting p by Corollary <ref>.
    The map
    (R;_p)→((R;_p)⊗_^)(R;_p)⊗_
    is an equivalence after inverting p. The induced map on the bottom horizontal
    fibers is
    Σ((R;_p)⊗_)_hS^1→Σ(R;_p)_hS^1,
    which is an equivalence after inverting p since
    (R;_p)⊗_→(R;_p) is an equivalence
	 after inverting p and this property is preserved by taking S^1-homotopy orbits. Thus, the bottom square is cartesian
    after inverting p. Using these identifications, the theorem follows.
 



Again, one can make effective the statement that (<ref>) is
cartesian after inverting p, at least in the range ≤ 2p-4. 
In this case, 
we find (via <Ref>) that 
for i ≤ p-1, τ_≤ 2i of the total cofiber of
(<ref>) is annihilated by p^i. Indeed, the map on cofibers of
the bottom rows of 
(<ref>) and 
(<ref>) is given by 
Σ^2 ((R) ⊗_ℤ^triv )_hS^1→Σ^2  (R; ℤ_p) _hS^1. This map is an equivalence in degrees
≤ 2p-2. 
 


 
Let R be a ring. 
Consider the map  (R ⊗_𝕊𝔽_p) →(R;
ℤ_p) from above. After inverting p, in view of <Ref>
below, 
we have an equivalence ( R ⊗_𝕊𝔽_p; ℚ_p)
≃(R/p; ℚ_p). 
We therefore obtain a map 
β(R/p; ℚ_p) →(R; ℚ_p), and precomposing with the
trace we obtain 

    tr_crys=tr∘β(R/p; ℚ_p) →(R; ℚ_p).
 
We call tr_crys the p-adic Chern character and record that it fits into a natural commutative diagram
    
    (R;ℚ_p)[r][d]_tr   (R/p;ℚ_p)[d]^tr@/^1.5cm/[dd]^tr_crys
    (R;ℚ_p)[r][d]    (R/p;ℚ_p)[d]^β
    ^-(R;ℚ_p)[r]    (R;ℚ_p)

in which the bottom square is a pullback.
 

 
In recent work, Petrov–Vologodsky <cit.> have shown that if 
p>2 and R is p-torsion free, then there is a natural equivalence
(R; ℤ_p) ≃(R/p; ℤ_p). 
Thus, one could attempt to compare the 
p-adic Chern character tr_crys with the usual trace
(R/p; ℤ_p) →(R/p; ℤ_p). We have not considered this
question. 
 

We can now give a quick proof of Theorem A by combining the above results with the following theorem.


If R is commutative and henselian along (p), then the trace induces an equivalence
(R,(p); _p) ≃(R,(p); _p).



 
If R is only associative, but p-complete and has bounded p-power
torsion,[If R is non-commutative and 
p-complete,  it is natural to ask whether there is still an
equivalence (R,(p); _p) ≃(R,(p); _p). We do not know the answer to this question.] then there is an equivalence
(R/p^n,
(p)) ≃(R,(p); _p).
This follows by the Dundas–Goodwillie–McCarthy theorem <cit.> and the
p-adic continuity of TC, cf. <cit.>. 
 


As we have already noted, the square (<ref>) from
    <Ref> is a pullback after inverting p; that is,
    the bottom square in (<ref>) is a pullback. But the top
    square in (<ref>) is a pullback by Theorem <ref>;
    assembling these cartesian squares completes the proof of the theorem.
 





 §.§ Fiber sequences up to quasi-isogeny


Next, we review some definitions and terminology as in <cit.>,
identify more carefully the fiber terms in the above squares, and prove
Corollary [quasiisogenythmB]B from the introduction.

 

Given an additive category (or ∞-category) 𝒞, we say that a
map f X → Y is an isogeny if there exists g Y → X and an
integer N > 0
such that g ∘ f = N id_X and f ∘ g = N id_Y. 
Let 𝒞 be a stable ∞-category equipped with a t-structure 
which is left-complete.[Recall that 𝒞 said to be
    left-complete (with respect to the given t-structure) if the natural map
→_n_≤ n is an equivalence. This is a technical
condition satisfied by many stable ∞-categories such as  and
().] We say that a map f X → Y of bounded below objects is 
a quasi-isogeny if the following equivalent conditions are satisfied: 


  * for each n, 
the map τ_≤ n fτ_≤ n X →τ_≤ n Y 
is an isogeny in 𝒞;

  * for each n, the map 
π_n X →π_n Y in the heart 𝒞^ is an isogeny. 


 
We will need some elementary observations about quasi-isogenies. 
A map f X→ Y of bounded below objects in  is a
quasi-isogeny if and only if the fiber fib(f) is quasi-isogenous to zero.
If one restricts to 𝒞_≥ 0 (i.e., connective
objects), then quasi-isogenies are preserved under finite colimits and
geometric realizations (but generally not under filtered colimits). 
Next, let 𝒞, 𝒟 be stable ∞-categories with left-complete
t-structures. 
Given a right t-exact functor F𝒞→𝒟 (or just a right
bounded exact functor), 
it is easy to see that F preserves quasi-isogenies.[A functor
→ is right t-exact with respect to fixed
t-structures on  and  if it restricts to a functor
_≥ 0→_≥ 0. It is right bounded if it
restricts to a functor _≥ 0→_≥ n for some n∈.]

Given an ∞-category ℐ, we will say that a natural
transformation f → g of functors 
f, g ℐ→𝒞 
is a quasi-isogeny if it is a quasi-isogeny in (ℐ, 𝒞)
with the pointwise t-structure. 
We will say that two functors are naturally quasi-isogenous if they are
are related by a zig-zag of quasi-isogenies of functors. 



For =, the map 𝕊→ℤ is a quasi-isogeny but
of course not an isogeny (as there is no nontrivial map back). In fact, in 
one has the following formality result of Beilinson <cit.>: every
bounded below spectrum X is quasi-isogenous to the spectrum ⊕_n
Hπ_n(X)[n]. In particular, two bounded-below spectra X and Y are
quasi-isogenous precisely if for each n separately the abelian groups π_nX
and π_n Y are isogenous. To see that every spectrum is formal in the above
sense, it suffices to observe that every k-invariant of a
connective spectrum X is
bounded torsion (where the torsion degree only depends on the degree of the
k-invariant and not on the specific homotopy groups). For explicit bounds,
cf. <cit.>. 

This formality result of course does depend on choices and thus does not give
similar results in functor categories 𝒞 = (ℐ ,). 


The fiber sequence of Corollary <ref> is the key to obtain our version of Beilinson's theorem 
<cit.>, as follows. 


For any associative ring R the following spectra are naturally quasi-isogenous
to each other (i.e., related via a natural zig-zag of quasi-isogenies)

    (R,(p);_p)         Σ(R, (p); _p)         Σ(R;_p)   .

Moreover,
    
        
  (a) 
if R is p-torsion free, then the first two are equivalent after (2p-5)-truncation, and

  (b)
if R is p-torsion free and[This is true pro-étale locally if R
is commutative, thanks to <cit.>.] π_-1( (R; ℤ_p)) = 0, then the
first two are equivalent after (2p-4)-truncation. 


 




For every associative ring R we have the following commutative diagram of fiber
sequences 

    Σ((R;_p) ⊗_ℤ_hC_p)_hS^1[r]   (R;_p)⊗_[r]   (R⊗__p;_p)
    
    F [u] [d] [r]    TC(R; ℤ_p) [d]^id[r]
    [u]    TC( R
    ⊗_𝕊𝔽_p; ℤ_p) [d] [u]^id
    (R,(p); _p)[r]   (R; _p)[r]   (R/p; _p).

To form the above diagram, 
we use the map  TC(R; ℤ_p) →TC(R; ℤ_p)
⊗_𝕊ℤ
induced from the map → as well as the map on TC(-;
ℤ_p) induced by the  Postnikov section R ⊗__p → R/p. All horizontal sequences in
(<ref>) are fiber sequences, either by <Ref> or by
definition; that is, F is defined as the fiber of TC(R;
ℤ_p) →TC( R ⊗_𝕊𝔽_p;
ℤ_p).

We claim now that all the vertical maps in diagram (<ref>)  are quasi-isogenies. 

 
The map 
TC(R; ℤ_p) →TC(R; ℤ_p)
⊗_𝕊ℤ in the 
diagram (<ref>) is a natural quasi-isogeny of spectra.
Moreover, its fiber is  (2p-4)-connective.
If π_-1((R; ℤ_p)) = 0, then the fiber is (2p-3)-connective.
 
 
The first part follows from the observation that tensoring a quasi-isogeny (in this case →) with a bounded below spectrum (here (R;_p)) is again a quasi-isogeny. 
Moreover, the fiber of (R; _p) →(R; _p)⊗_ is (2p-4)
connective since (R;_p) is (-1)-connective and the fiber of _(p)→_(p) is (2p-3) connective. 
The last assertion follows similarly. 
 

The right horizontal map (R⊗__p;_p)→(R/p;_p) in diagram (<ref>) is also a quasi-isogeny. This
follows from <Ref> that we will discuss and prove in
Section <ref> and which is purely internal to cyclotomic spectra. But
we also want to give a direct proof here using -theory and the
Dundas–Goodwillie–McCarthy theorem. 


The natural map (R ⊗__p;_p) →(R/p; _p) is a quasi-isogeny. If R is p-torsion free then the fiber is (2p-1)-connective.


The map of connective ring spectra R ⊗__p → R/p is an
isomorphism on π_0. Thus the Dundas–Goodwillie–McCarthy theorem (for ring spectra) implies that its fiber is equivalent to the fiber of the map

    (R ⊗__p;_p) →(R/p; _p)

But the map R ⊗__p → R/p of ring spectra is a quasi-isogeny
and, if R is p-torsion free, has fiber which is (2p-2)-connective. Thus, the
map on -theory is a quasi-isogeny and has fiber which is (2p-1)-connective,
cf. <cit.> (the proof in loc. cit. shows that the map is truly a
quasi-isogeny of functors).  


Now we know that the vertical maps in diagram (<ref>) are quasi-isogenies, so we conclude that Σ((R,_p) ⊗_ℤ_hC_p)_hS^1
and (R, (p); _p) are quasi-isogenous to one another. Moreover, if R is
p-torsion free, then the vertical maps from F have (2p-4)-connective
fibers by the above discussion (which upgrades to (2p-3)-connective fibers if
π_-1(R; ℤ_p) = 0). Thus, we conclude that  Σ((R,_p) ⊗_ℤ_hC_p)_hS^1
and (R, (p); _p) are equivalent in degrees ≤ (2p-5), and in
degrees ≤ 2p-4 if π_-1(R; ℤ_p) = 0. 
<Ref> now follows from the arguments above and the following lemma. 
 



The following spectra are naturally quasi-isogenous to each other

    ((R;_p) ⊗_ℤ_hC_p)_hS^1        (R, (p); _p)         (R;_p)

and the first two are equivalent after (2p-4)-truncation if R is p-torsion free.


We have that (R;_p) ⊗_ℤ_hC_p is equivalent to the fiber of 

    (R; _p)⊗_→(R ⊗__p; _p)

and this map sits in a commutative square 

    (R; _p)⊗_[r] [d]   (R ⊗__p; _p)  [d]
    (R; _p) [r]    (R/p; _p)

of spectra with S^1-action. Both vertical maps are quasi-isogenies, so that we
get the desired quasi-isogeny between the first two terms of the statement by
taking S^1-orbits. The term (R/p) is 
quasi-isogenous to 0, so that we get the last quasi-isogeny too. 
If R is p-torsion free, the fibers of the left and right vertical maps are in
degrees ≥ 2p-3 and ≥ 2p-2, respectively, so the last assertion follows
too. 




Corollary [quasiisogenythmB]B follows by
combining Theorem <ref> with Theorem <ref>.  In particular, we
have an isomorphism _*(R, (p);_p) _*-1(R, (p);_p) for
∗≤ 2p-5, for R commutative, p-torsion free, and henselian along
(p).
Note also that 
with the same proof,  we can deduce the following variant of Theorem
<ref> for arbitrary connective ℤ-algebra ring spectra
(also known as -linear dgas).


    If R is a connective -algebra spectrum, then the fiber of (R;_p)
→(R ⊗__p; _p) is quasi-isogenous to Σ(R;
ℤ_p) and after (2p-5)-truncation equivalent to the fiber of

    Σ(R;_p) →Σ(R ⊗__p; _p)  .





 
In all of the above, the denominators involved in the above quasi-isogenies are
uniform: they do not depend on the choice of R. More formally, one could state
all of the above quasi-isogenies via the ∞-category of functors from
rings R to spectra. The denominators in the next result are not independent in the same fashion. 
 

 
Let (R, I) be a pair consisting of an associative ring R and a nilpotent ideal I. 
Then there is a natural zig-zag of quasi-isogenies between (R, I;
ℤ_p) and Σ(R, I; ℤ_p). 
 
 
By the Dundas–Goodwillie–McCarthy theorem, we can replace -theory with . 
We have a natural map 
    
    (R, I;ℤ_p) →fib( (R, (p); ℤ_p) →(R/I, (p);
    ℤ_p))

Now (R/p; ℤ_p) →(R/(I, p); ℤ_p)
is a quasi-isogeny in view of <Ref> below, 
so that (<ref>) is a quasi-isogeny. Combining with the quasi-isogenies
of <Ref> now completes the proof. 
 





§ QUASI-ISOGENIES OF CYCLOTOMIC SPECTRA


In this section, we systematically study quasi-isogenies in  cyclotomic spectra,
give another
proof of Theorem [thm:a]A and
Corollary [quasiisogenythmB]B, and prove Theorem [thm:c]C,
sharpening some
results of Geisser–Hesselholt <cit.>.




 §.§ Preliminaries

We will apply the notion of quasi-isogeny (<Ref>) to the
∞-category  of  cyclotomic
spectra using the t-structure of <cit.>;[Recall that what we
denote by  is denoted _p in <cit.>.] 
this t-structure is defined so that the connective objects of  are
those whose underlying spectrum is connective and it is checked
in <cit.> that the t-structure is left-complete. 
Note that a quasi-isogeny of bounded-below cyclotomic spectra f X → Y is a quasi-isogeny of
underlying spectra, and (f;_p)(X;_p) →(Y;_p) is also a quasi-isogeny. However,  (𝔽_p)
∈ has underlying spectrum quasi-isogenous to zero, but is not itself
quasi-isogenous to zero because (𝔽_p)(_p;_p) is torsion free and
nonzero: π_0(_p;_p)π_-1(_p;_p)_p.


In the next result, we use the notion of  of a cyclotomic spectrum, which
plays an important role in the work <cit.>. See <cit.> for an account
of  in the approach to cyclotomic spectra via genuine equivariant homotopy theory. 
Implicitly,  is computed with respect to our fixed prime p, but it will
not generally be p-complete unless we p-complete it forming (X;_p) for a
cyclotomic spectrum X.


A map f X → Y of bounded-below cyclotomic spectra is a
quasi-isogeny in 
if and only if the map of spectra (f)(X) →(Y) is a quasi-isogeny
of spectra.
 
 
    This follows from the description of 
    the cyclotomic t-structure of <cit.>. In particular, the cyclotomic homotopy groups
    of X ∈ are precisely the homotopy groups of the spectrum (X),
    together with the Frobenius and Verschiebung maps. These are p-typical
    Cartier modules, which is to say abelian groups M with endomorphisms F and V
    satisfying FV=p,
    and the heart ^ is equivalent to a full subcategory of the
    category of p-typical Cartier modules. Now, it suffices to check that a
    map h W→ Z between p-typical Cartier
    modules is an isogeny precisely if the underlying map of abelian groups is
    an isogeny. One implication is immediate and for the other suppose
    g Z→ W is a map of abelian groups such that g∘ h=Nid_W and h∘
    g=Nid_Z. The map g might not respect the F and V maps.
    However, for z∈ Z, F(g(z))-g(F(z)) and V(g(z))-g(F(z)) are both in
    the kernel of h, which consists of N-torsion elements of W.
    Therefore, Ng is a map of p-typical Cartier modules, Ng∘
    h=N^2id_W, and h∘ Ng =N^2id_Z.
 

 
Let X ∈ be a cyclotomic spectrum such that X is bounded-below, such
that the
Frobenius φ X → X^tC_p is nullhomotopic in (BS^1, ), and 
such that X is quasi-isogenous to zero as a spectrum. 
Then X is quasi-isogenous to zero as a cyclotomic spectrum. 


 
The assumption that the Frobenius is nullhomotopic implies that TR(X)
is a product ∏_n ≥ 0 X_hC_p^n, using the description of  as an
iterated pullback, cf. <cit.> and <cit.>. 
The assumption that X is quasi-isogenous to zero now implies that the
above product 
    is also quasi-isogenous to zero, so we conclude by
    Proposition <ref>. 
 

We observe that the theory of cyclotomic spectra 
admits a natural graded variant. 
A graded spectrum is an object of the functor category (ℤ_≥ 0^ds,
) where ℤ_≥ 0^ds denotes the discrete category
of nonnegative integers with no non-identity morphisms; given a graded spectrum X we let X_i ∈, i
≥ 0 denote the ith graded piece. 
We let GrSp denote the ∞-category of graded spectra, which we
consider as a symmetric monoidal ∞-category under Day convolution using
the multiplication symmetric monoidal structure on ℤ_≥
0^ds. 
A graded cyclotomic spectrum X consists of a 
graded 
spectrum X = {X_i} equipped with an S^1-action together with a
family of S^1-equivariant maps 
φ_i X_i → X_pi^tC_p for i ≥ 0.  
We let  denote the ∞-category of graded cyclotomic spectra. 
Any graded cyclotomic spectrum X = {X_i} has an
underlying cyclotomic spectrum 
⊕_i ≥ 0 X_i, and this defines a forgetful functor →. 

More formally, the ∞-category  is defined  as follows.  We consider the ∞-category 
(BS^1, GrSp) 
of graded spectra equipped with an S^1-action. 
This admits a natural endofunctor F
which sends {X_i, i ≥ 0} to {X_pi^tC_p}, 
where we regard X_pi^tC_p as a spectrum with an S^1/C_p ≃
S^1-action. Then  is defined as the ∞-category of
F-coalgebras, as in <cit.>. 

Given a graded ring spectrum R, there is a graded  cyclotomic spectrum 
(R) 
obtained by applying the cyclic bar construction in the category of graded
spectra. This refines the usual  and admits an S^1-action in graded
spectra. See <Ref> for the details of this construction. Compare also <cit.> for a treatment of filtered cyclotomic spectra and
filtered  using more classical methods. 


 

Let X be a graded  cyclotomic spectrum.
If
    
        
  (1) the underlying spectrum of X is quasi-isogenous to zero,
        
  (2) the graded piece X_0 is contractible, and
        
  (3) the connectivity of the pieces X_i tends to infinity in i,

then X is quasi-isogenous to zero as an object of . 
 
 
Given a graded cyclotomic spectrum X, for each i, we can
construct a graded 
cyclotomic spectrum X_≤ i∈  such that (X_≤ i)_j = 0 for
j > i and (X_≤ i)_j = X_i for j ≤ i and a tower of maps 
X →…→ X_≤ n→ X_≤ n-1→…→ X_≤ 1. 
This is a tower in , and we can consider it as a tower of underlying
objects in  too. 

We need to show that for each j, π_j( (X)) is isogenous to zero. 
Our assumptions imply that π_j((X)) →π_j  ( X_≤ n) is an
isomorphism for n ≫ 0.
However, the object fib( X_≤ i→ X_≤ i-1) defines a
cyclotomic spectrum with Frobenius homotopic to zero, in view of the grading. 
It follows from <Ref> that 
( fib(X_≤ i→ X_≤ i-1)) is quasi-isogenous to zero, and by
induction ( X_≤ n) is quasi-isogenous to zero. Putting these observations
    together with Proposition <ref> completes the proof. 
 





 §.§ Quasi-isogenies on 

Our main result here is the following, which restates
Theorem [thm:c]C.
On   and for discrete rings in which p is nilpotent, it is due to Geisser–Hesselholt
<cit.>, and the main arguments are based on theirs. 

 

Let f A → A' be a map of connective associative ring spectra. 
If

    
  (i) f is a quasi-isogeny of spectra and
    
  (ii) the map π_0(f)π_0(A) →π_0(A') is surjective with nilpotent
kernel,

then (A) →(A') is a quasi-isogeny in . 
 


 We will first verify some special cases. 

 

Let R be a connective associative graded ring spectrum. 
If

    
  (a) each R_i, i > 0, is isogenous to zero as a spectrum and
    
  (b) the connectivity of the R_i tends to ∞  as i →∞,

then the map (R) →(R_0) is a quasi-isogeny in
.   
 
Since R is a graded ring spectrum, (R) admits the structure
of a graded cyclotomic spectrum (refining the usual cyclotomic structure on
(R)), and in degree zero one has (R_0). 
For this, compare <Ref>, or the work of Brun <cit.>, who uses 
the more classical approach to cyclotomic spectra. 

Now we wish to apply <Ref>. 
Consider the subcategory 𝒞⊆GrSp_≥ 0 of connective graded spectra 
spanned by graded spectra Z such that Z_i is quasi-isogenous to zero for i
>0 and such that the connectivity of Z_i grows without bound as i →∞. Then 𝒞 is closed under tensor products and geometric
realizations. 
The assumptions on R imply that R ∈𝒞, and
consequently 
(R) ∈𝒞 as well. That is, 
(R)_i is quasi-isogenous to zero for i
>0 and the connectivity of (R)_i grows without bound as i →∞. 
Thus, we can apply 
<Ref>. 
 


 

Let A be a connective associative ring spectrum. 
Let M be a connective (A, A)-bimodule which is quasi-isogenous to zero. 
Suppose A is a square-zero extension of A by M, in the sense
that 
one has a map 
f A → A ⊕ M[1] in _/A and a pullback diagram

    A[d]  [r]      A [d]^0  
    
    A [r]^-f    A ⊕ M[1].

Then the map (A) →(A) is a quasi-isogeny in . 
 
 
We can form the Čech nerve of the map A→ A, i.e., the
simplicial object
…A×_A A⇉A. 
This yields a simplicial object X_∙ of  which resolves A. 
It follows that |(X_∙)| ≃(A) in . 

Now A×_A A is a trivial square-zero extension of
A by M. It follows 
that (A×_A A) is quasi-isogenous to
(A) by <Ref>, since a trivial square-zero extension can be
given a grading. Continuing in this way, it follows that all the 
maps in the simplicial object (X_∙) are quasi-isogenies. 
Taking geometric realizations now, it follows that
(A) → |(X_∙)| ≃(A) is a quasi-isogeny in
. 
 



 

Let B be a connective associative ring spectrum and let 
B' be 
an object of _B//B. Suppose that 
the augmentation map B' → B is a quasi-isogeny  and the map π_0(B') →π_0(B) has nilpotent kernel. Then (B) →(B') is a quasi-isogeny.
 
 
Recall first that _B//B is equivalent to the ∞-category of
nonunital associative algebra objects in (B, B)-bimodules. 
In particular, I = fib(B' → B) has such a structure.
We can work up the Postnikov tower τ_≤⋆ I; since
 behaves well with respect to Postnikov towers, it suffices to prove the
result for each τ_≤ n I. 
General results as in <cit.> (which go back at least to
<cit.>) 
now show that τ_≤ n I can be obtained in finitely
many steps via square-zero extensions from B, by bimodules which 
are quasi-isogenous to zero.
Now we conclude via <Ref>. 
 

 
We consider the Čech nerve of A → A'. 
We obtain a simplicial object X_∙ in _≥ 0 
such that |X_∙| ≃ A' and such that each X_i is an iterated fiber
product of copies of A over A'. 
Each X_i can be given the structure of an object of _A//A 
(via appropriate face and degeneracy maps), 
whence we conclude by <Ref> that all the maps in the simplicial object
(X_∙) are quasi-isogenies in . Finally, the result now
follows by taking geometric realizations. 
 

One important corollary of Theorem <ref> is the following result
of Geisser and Hesselholt; see <cit.> for generalizations.

 
    If p is nilpotent in A and I⊆ A is a two-sided
    nilpotent ideal, then (A,I) ≃(A, I) is quasi-isogenous to zero.



    In this case, A→ A/I is a quasi-isogeny (they are both
    quasi-isogenous to zero) and hence Theorem <ref> applies to
    prove that (A)→(A/I) is a quasi-isogeny of cyclotomic
    spectra. This implies in particular that (A,I) is quasi-isogenous to
    zero. 






 §.§ Quasi-isogenies and the Beilinson fiber sequence


Recall that Proposition <ref>, which was key in proving Theorem [thm:a]A, asserts that for every ring R the induced map 
 (R ⊗__p;_p) →(R/p; _p) is a quasi-isogeny. Our
 proof in Section <ref> relied on -theory. Alternatively we can now
 also deduce this fact directly from Theorem <ref>, which implies
 that (R ⊗__p) →(R/p) is a quasi-isogeny of cyclotomic
 spectra and therefore (R ⊗__p;_p) →(R/p; _p) is a
 quasi-isogeny of spectra. 
 This immediately implies also the following variant of 
Theorem [thm:a]A. 

 
Let R be a ring. Then there is a natural pullback square

    (R; ℚ_p) [d]  [r]    (R/p; ℚ_p) [d]  
    ^-(R; ℚ_p) [r]    (R; ℚ_p).
 
 


 In this section we want to take this a step further and prove a cyclotomic version of Theorem [thm:a]A.


For every ring R, the following  cyclotomic spectra are quasi-isogenous to each other

    (R, (p);_p)         (R;_p)          (R,
        (p);_p)

where  (R;ℤ_p) and (R, (p);ℤ_p) are equipped with the canonical S^1-actions and  the zero Frobenius (see Example <ref>). 
Moreover if R is p-torsion free then we have an equivalence of cyclotomic spectra

    τ^cyc_≤ (2p -4)(R, (p);_p)   ≃  τ^cyc_≤ (2p - 4)(R, (p);_p).




We note that the last theorem immediately implies Theorem [thm:a]A by passing to (-; _p) since (-;_p) of a cyclotomic spectrum with zero Frobenius is just given by the p-completion of the shifted S^1-orbits. But Theorem <ref> is strictly stronger than Theorem [thm:a]A since quasi-isogenies cannot be detected on . The remainder of this section is devoted to proving Theorem <ref>. 

    Suppose that X→ Y is a quasi-isogeny of  cyclotomic spectra and
    that M is any bounded below cyclotomic spectrum. Then, X⊗_
    M→ Y⊗_ M is a quasi-isogeny of cyclotomic spectra.



    We can assume M is connective, in which case the functor -⊗_ M is
    a right t-exact endofunctor of  and hence preserves quasi-isogenies.


We now want to apply a similar proof-strategy as in Section <ref> and consider the diagram

    (R;
    ℤ_p)⊗__hC_p[r]   (R;
    ℤ_p)⊗_^[r]   (R⊗__p;
    ℤ_p)[d]
    (R, (p); ℤ_p)[r]   (R; ℤ_p)[u][r]   (R/p;
    ℤ_p),

of cyclotomic spectra in which the horizontal rows are fiber sequences. 
Both vertical maps are quasi-isogenies in cyclotomic spectra: the first
by Lemma <ref> and because ^→^ is a quasi-isogeny (since (-)^
is right t-exact) and the second by Theorem <ref>.
The right vertical map has homotopy fiber in degrees ≥ 2p-2, while the
middle vertical map has homotopy fiber in degrees ≥ 2p-3. 
It follows that (R, (p); _p) is quasi-isogenous in cyclotomic spectra to
(R; ℤ_p)⊗__hC_p, and their cyclotomic
(2p-4)-truncations τ_≤ 2p-4^cyc are equivalent. Note that the cyclotomic
Frobenius on (R;_p)⊗__hC_p is nullhomotopic by the Tate
orbit lemma.



Now the next lemma finishes the proof of Theorem <ref>.


The following cyclotomic spectra are quasi-isogenous to each other

    (R; ℤ_p)⊗__hC_p        (R;_p)
                (R, (p);_p).

Moreover, the cyclotomic truncations τ_≤ 2p-4^cyc of the
first and the third are
naturally equivalent. 


We consider the square

    (R; _p)⊗_[r] [d]   (R ⊗__p; _p)  [d]
    (R; _p) [r]    (R/p; _p)

of spectra with S^1-action, in which the vertical maps are quasi-isogenies
    and the right hand terms are quasi-isogenous to zero. We consider it as a
    square of cyclotomic spectra by equipping all spectra with the zero
    Frobenius map. It follows from Proposition <ref> that the
    vertical maps are quasi-isogenies of cyclotomic spectra. The horizontal fibers are
    equivalent to (R;_p)⊗__hC_p and (R, (p);_p),
    which shows they are quasi-isogenous to each other.

Finally, the induced map of cyclotomic spectra
(with zero Frobenii)
(R;_p)⊗__hC_p→(R, (p);_p)
has the property that it is an equivalence of underlying spectra in degrees ≤ 2p-4
(as in <Ref>) 
and
consequently induces an equivalence on 
cyclotomic homotopy groups in degrees ≤ 2p-4, e.g., using the description
of  as in the proof of <Ref>. 










§ APPLICATION TO P-ADIC DEFORMATIONS



In this section, we prove
Theorems [thm:d]D and [thm:e]E.
Throughout this section, let 𝔛 be a quasi-compact and
quasi-separated (qcqs) p-adic formal scheme  with bounded
p-power torsion, and write 𝔛_n=𝔛×__p/p^n. 
We are interested in the following invariants of , and in
particular the p-adic deformation problem (<Ref> below).   


Let F be an invariant of schemes (such as , , , 
^-, ,).
Given the formal scheme , we define
F^cts(𝔛)  via

    F^cts( 𝔛) = _n F(𝔛_n)
    .
 
 

 If the p-adic formal scheme  arises as the p-adic
completion of a scheme X, we have a natural comparison map 

    F( X) → F^cts( ) .


 

Suppose  is the p-adic completion of a qcqs scheme X with bounded p-power
torsion. Then the maps 
(<ref>) for F = , , ^-, , are p-adic equivalences. 
 
 
Using Zariski descent on X, we may assume that X = (R) where R is a
ring of bounded p-power torsion, and then = ( R̂_p). 
Using the cyclic bar construction, 
it is not difficult to show that 
^cts( ; ℤ_p) = (R; ℤ_p), i.e., that 
(<ref>) is a p-adic equivalence for F =
(cf. the
proof of <cit.>). 
Tensoring over (ℤ) with ℤ, one deduces the result for
, and then
taking S^1-invariants and coinvariants, we find that 
(<ref>) is a p-adic equivalence for F = ^-,. 
Running the above argument with  instead of , one concludes 
that (<ref>) is a p-adic equivalence for F =
<cit.>. See also <cit.> for these results,
when R is assumed noetherian and F-finite. 
 



By contrast, it is much more difficult to control (<ref>) when F =. We mention the two following cases. 

[Formal affine schemes] 
Suppose 𝔛 is affine, i.e., 
𝔛 = (R), for R a p-adically complete ring with bounded
p-power torsion. We can then write 𝔛 as the p-adic completion
(as a formal scheme) of X = (R). 
In this case, 
the comparison map 
(<ref>) is a p-adic equivalence for F = as well, cf. 
<cit.> and <cit.>. 

[Proper schemes]
Suppose R is  p-complete. 
Suppose 𝔛 is the p-completion of a proper scheme X
→(R). The map (X; ℤ_p) →^cts(𝔛;
ℤ_p) is probably not an equivalence; compare <cit.> for
a related counterexample in equal characteristic zero. 
In this case, ^cts( 𝔛; ℤ_p) is generally much
more tractable than (X; ℤ_p) via comparisons with topological
cyclic homology. 
 



[The p-adic deformation problem] 
Let 𝔛 be a p-adic formal scheme with special fiber 𝔛_1 as above.

For i ≥ 0,
what is the image[By the Milnor exact sequence, this is equivalent to describing the image of the map 
(_n _i(𝔛_n)))_ℚ→_i(𝔛_1; ℚ).] 
of the map

    _i^cts(𝔛; )→_i(𝔛_1; ) ?

 

We first observe that <Ref> is essentially a p-adic question in
. 
For each n ≥ 1, let  (_n,_1) be the
fiber of (_n)→(_1). Since _n is a p-adic nilpotent thickening of _1, 
the relative -theory (_n, _1) has homotopy groups which are
bounded p-power torsion (cf. <Ref>, due to <cit.>), and the spectrum is therefore p-complete. 
Using the Dundas-Goodwillie-McCarthy theorem <cit.>, and taking limits, we obtain a
cartesian square

    ^cts(
    𝔛) [r][d]   (_1)[d]
    ^cts( 𝔛; ℤ_p)[r]   (_1;_p).

Since the _n, n ≥ 1 are p-power torsion schemes, their  are already p-adically complete. 
Using this diagram, we see that  it suffices to determine the image of the map 
_i^cts( 𝔛; ℚ_p) →_i( 𝔛_1;
ℚ_p). 

In this section, we will describe an explicit obstruction class for
Question <ref> in case i = 0 
(sharpening results of <cit.>) in certain geometric situations and construct general obstruction 
classes in all cases (after <cit.>). 



 §.§ The Bloch–Esnault–Kerz theorem


In <cit.>, Bloch–Esnault–Kerz consider Question <ref> in the
case i = 0 and where 𝔛 has the following form. 
Let K be a complete discretely valued field of
characteristic zero with ring of integers 𝒪_K,
whose residue field k is perfect of characteristic p>0. We let π∈𝒪_K be a uniformizer and denote by K_0 the ring of fractions
W(k)[1p]. 
We take 𝔛→Spf(𝒪_K) to be a smooth p-adic formal scheme, 
with special fiber 𝔛_k →(k) and rigid analytic generic
fiber 𝔛_K
over K. The goal is to understand the image of the map 
_0^cts( 𝔛; ℚ_p) →_0(𝔛_k; ℚ_p). 

We refer to <cit.> for more detailed motivation for the above question, as
well as <cit.> for discussions of the analogous question in
equal characteristic. 
Note that when 𝔛 arises from a smooth proper scheme X →
(𝒪_K), the above question says nothing about the image of the map 
_0(X; ℚ)→_0(X_k; ℚ); this (at least up to homological
equivalence) is the subject of the far more difficult p-adic variational Hodge conjecture 
of Fontaine–Messing 
(Conjecture <ref>). 



Here we will unwind the Beilinson fiber square to answer
Question <ref> in this case in terms of the crystalline Chern
character. 
To begin with, we need to review the crystalline Chern
character and the crystalline to de Rham comparison. 


[de Rham cohomology] 
Given a smooth p-adic formal scheme →Spf(𝒪_K), 
we will consider the (p-adic)
de Rham cohomology R Γ_(/𝒪_K) ∈
D(𝒪_K), equipped with the
descending, multiplicative Hodge
filtration 
Fil^≥⋆ R Γ_(/𝒪_K). 
When  is also assumed proper, then 
all of these are perfect complexes in D(𝒪_K). 
Furthermore, after inverting p, we write
R Γ_(_K/K) ∈ D(K) and 
Fil^≥⋆ R Γ_(_K/K) for the induced objects. 

A basic fact we will use is that when  is proper, the induced spectral sequence from the Hodge
filtration 
on R Γ_(_K/K)
degenerates after rationalization; this is the degeneration of the
Hodge-to-de Rham spectral sequence for proper smooth rigid analytic varieties,
proved by Scholze <cit.>. 
 

[Comparison between crystalline and de Rham cohomology] 

Given →(𝒪_K) a smooth p-adic formal scheme, 
we can consider the crystalline cohomology R Γ_(_k) of the
special fiber as well. In the absolutely unramified case (when 𝒪_K = W(k)), the usual de
Rham to crystalline comparison theorem yields an equivalence 
R Γ_(_k) ≃ R Γ_(/𝒪_K). 
In general, by <cit.>, we have a natural equivalence after
rationalization

    R Γ_(_K/K) ≃  R
    Γ_(_k; ℚ_p)  ⊗_K_0 K.

 
[The crystalline Chern character] 
Let Y be a regular scheme of 
characteristic p. 
Given a vector bundle 𝒱 on Y, we can define Chern classes
c_i (𝒱) ∈ H^2i_crys(Y) for i ≥ 0 satisfying the
usual axioms, cf. <cit.> (e.g., using the classical method of <cit.>). The usual formula then yields a crystalline Chern
character, i.e., a
natural ring homomorphism
into the rationalized crystalline cohomology

    ch_crys_0(Y) →⊕_i ≥ 0
    H^2i_crys(Y; ℚ_p),

which carries the class of a  line bundle  ℒ to 1 +
c_1(ℒ). 
 


Our main result is the following theorem, which extends results of
Bloch–Esnault–Kerz <cit.>. In <cit.>, this result is proved in the
case where K=K_0 is absolutely unramified, X arises
from a smooth projective scheme, and  p >
dim(X) + 6. In <cit.>, it is shown that there is an obstruction 
in ⊕_i ≥ 0 H^2i_dR(X_K)/Fil^≥
iH^2i_dR(X_K), but the obstruction is not identified with the
Chern character; see Section <ref> below for more discussion. 

 

Let K be a complete discretely valued field of characteristic zero with ring of integers 𝒪_K,
whose residue field k is perfect of characteristic p > 0. Let →Spf(𝒪_K) be a
 proper smooth p-adic formal scheme with special fiber _k.
A class x ∈_0(_k; ℚ_p) lifts to
_0^cts(;
ℚ_p) if and only if the
crystalline 
Chern character ch_crys(x) ∈⊕_i ≥ 0
H^2i_crys(_k; ℚ_p) maps (via the comparison map of
(<ref>)) to 
⊕_i ≥ 0Fil^≥ i H^2i_(_K/K) ⊆⊕_i≥ 0
H^2i_(_K/K). 
 

The proof of <Ref> will be carried out as follows. First, we give an
analogous form of the Beilinson fiber square when we work relative to
𝒪_K (<Ref>). Next, we will show that 
the p-adic Chern character can be defined entirely in terms of the
special fiber (which will use some Kan extension techniques from
<Ref>), and then identify it with the crystalline Chern character
(<Ref>). <Ref> will then follow directly.  

In the next result, we 
will use the continuous Hochschild (resp. negative cyclic, periodic cyclic)
homology of a formal scheme over 𝒪_K, defined as 
in <Ref>; note that <Ref> applies to these
relative theories too, since they can be recovered from .  

 

Let  
be a smooth formal 𝒪_K-scheme. Then there are natural fiber
squares
 
    ^cts(; ℚ_p) [d]  [r]    (_k; ℚ_p) [d]
    
    ^cts(; ℚ_p) [d]  [r]    (_k; ℚ_p)
    [d]  
    ^-, cts(/𝒪_K; ℚ_p) [r]    ^cts(/𝒪_K; ℚ_p).

 
 
This will follow from the Beilinson fiber square. 
By Zariski descent of all terms in 
the formal scheme , we can assume that = (R) for R a formally
smooth, p-complete 𝒪_K-algebra. 
First, 
( 𝒪_K; ℤ_p) ≃( 𝒪_K/W(k);
ℤ_p). Since
L_𝒪_K/W(k) is quasi-isogenous to zero, we find that  
the map 
( 𝒪_K; ℚ_p) → K given by truncation
is an equivalence. 
We thus conclude (via Hochschild–Kostant–Rosenberg) ^cts(R; ℚ_p) →^cts(R/𝒪_K; ℚ_p) 
is an equivalence, whence (R; ℚ_p) →(R/𝒪_K;
ℚ_p) is an equivalence too by taking S^1-coinvariants. 
Here we use that 
p-adic completion commutes with taking S^1-coinvariants on connective
spectra, since taking S^1-coinvariants behaves as a finite colimit in any
ranges of degrees. 
Therefore, the diagram

    ^-(R; ℚ_p) [d]  [r]    (R; ℚ_p) [d]  
    ^-(R/𝒪_K ; ℚ_p) [r]    (R/𝒪_K; ℚ_p)

is homotopy cartesian. Combining with the Beilinson fiber square, the
result now follows. 
 

[The p-adic Chern character map] 
Since /𝒪_K is smooth, we obtain from 
Hochschild–Kostant–Rosenberg type filtrations (as in <cit.>,
using Adams operations as in <cit.> to split the filtration)
natural decompositions

    ^cts(/𝒪_K; ℚ_p) ≃∏_i ∈ℤ 
    R Γ_(_K/K)[2i]
 
and 
    ^-,cts(/𝒪_K; ℚ_p) ≃∏_i ∈ℤFil^≥ i R Γ_dR(_K/K)[2i].

It follows that we obtain from (<ref>) a natural map 

    (_k; ℚ_p) →(_k;
    ℚ_p) →∏_i ∈ℤ R
    Γ_(_K/K)[2i]

for every smooth p-adic formal scheme →(𝒪_K). 
We observe that both the source and target actually depend only on the special
fiber _k of , thanks to <Ref>. 
Furthermore, to construct 
(<ref>), it suffices to work with affine formal schemes over
𝒪_K, by Zariski descent of the target, so we can assume 
= Spf(R) for R a formally smooth 𝒪_K-algebra. 
That is, we have a natural  map (R ⊗_𝒪_K k; ℚ_p) →∏_i ∈ℤ (R
Γ_crys( Spec(R ⊗_𝒪_K
k);ℚ_p) ⊗_K_0 K)[2i]. 


Consider the functors
on smooth k-algebras 

    A ↦(A; ℚ_p)  and  
    A↦∏_i ∈ℤ( RΓ_crys( (A);
    ℚ_p) ⊗_K_0 K )[2i].

The left Kan extension of (-; ℚ_p) to 
almost finitely presented objects of SCR_k (as in <Ref>)
is (-; ℚ_p) again, since this functor commutes with 
geometric realizations in SCR_k. By 
<Ref>, (-; ℚ_p) = ( π_0(-); ℚ_p) on
    SCR_k, so hypotheses (1) and (2) of
<Ref> are satisfied when applied to the left Kan extensions
of the functors (<ref>) on smooth k-algebras
(and Zariski sheafifying again). It follows that (<ref>) actually upgrades to a natural
transformation of functors in the special fiber alone. That is, for 
every smooth k-scheme Z, we obtain a natural map 

    (Z; ℚ_p) (Z; ℚ_p) →∏_i ∈ℤ(  R
    Γ_(Z; ℚ_p) ⊗_K_0 K)[2i],
 
such that (<ref>) is obtained by taking Z = X_k. 


 
Next, we identify (up to scaling factors) the map (<ref>) on π_0 with
the crystalline Chern character.

 

There exists a scalar λ∈ K^× such that 
for every smooth separated k-scheme Z, the map 
_0(Z; ℚ_p) →∏_i ≥ 0 H^2i_(Z;_p)
⊗_K_0 K of
(<ref>) is given by 
the crystalline Chern character composed with the automorphism that multiplies
the ith factor by λ^i. 
 
 
 It suffices (by the
resolution property) to evaluate 
(<ref>) on the class of a vector bundle on Z, and for this we will
reduce to the universal case. 
For this it
will be convenient to extend to stacks over k as well.
Given a smooth scheme or stack Z over k, 
let Vect_n(Z) denote the groupoid of n-dimensional vector bundles
on Z. 
It follows that we obtain from (<ref>) a natural transformation (of spaces)
for all smooth k-schemes Z,

    f_nVect_n(Z) →Ω^∞( ∏_i ∈ℤ(R
    Γ_(Z; ℚ_p) ⊗_K_0 K )[2i] )

such that the {f_n} are additive and  multiplicative. 

Both the source and target of the f_n are sheaves of spaces for the smooth or
étale topology on smooth k-schemes. 
Sheafifying for the smooth topology, we obtain such a natural transformation for
any smooth Artin stack, which still satisfies the additivity and
multiplicativity properties. 
By naturality, it suffices to show that 
for Z = BGL_n and for ℰ the tautological n-dimensional vector
bundle, f_n(ℰ) is given by (up to scalars) the crystalline Chern
character of ℰ. 
It follows that for each n, f_n(ℰ) is given by a power series
(with K coefficients) in the 
crystalline 
Chern classes of ℰ, since 
RΓ_crys(BGL_n; ℚ_p) (defined via sheafification) is the polynomial ring
K_0[c_1, …, c_n], e.g., as in the calculations of de Rham and
Hodge cohomology of BGL_n in <cit.>. 
By additivity, multiplicativity, and the splitting principle to reduce to the
case of line bundles, we find easily that f_n must be the Chern
character up to normalization by powers of some constant λ. 
Moreover, λ≠ 0 by comparison with the left-hand-side of
(<ref>). 
 



 
We use the fiber square of (<ref>). 
As before, we have identifications
^cts(𝔛/𝒪_K; ℚ_p) ≃∏_i ∈ℤ R
Γ_(𝔛_K/K)[2i] and 
^-,cts(𝔛/𝒪_K; ℚ_p) ≃∏_i ∈ℤFil^≥ i R Γ_(𝔛_K/K)[2i]. 
Using the crystalline-to-de Rham comparison (<Ref>) and
<Ref>, we see that the map 

    _0(𝔛_k; ℚ_p) →∏_i ∈ℤ H^2i_(𝔛_K/K) ≃∏_i ∈ℤ H^2i_(𝔛_k; ℚ_p) ⊗_K_0 K

is given up to scalar factors by the crystalline Chern character. 
The result now follows from 
<Ref>. 
 



 §.§ Generalization of Beilinson's obstruction; proof of
Theorem E



Let K, 𝒪_K, k be as in the preceding subsection. 
Consider a proper 
scheme X →(𝒪_K) with smooth generic fiber X_K and
possibly singular special fiber X_k. 
In <cit.>, Beilinson 
considers more generally the 
deformation problem for classes in higher -theory, and proves: 

 

Given x ∈_i(X_k)_ℚ, there is a natural  obstruction class in 
⊕_r ≥ 0 H^2r-i_dR(X_K/K)/ Fil^r
H^2r-i_dR(X_K/K) which vanishes
if and only if x lifts to (_i(X/π^n))_ℚ. 
More precisely, there is a natural equivalence of spectra

    cofib( ^cts(X; ℚ_p) →(X_k;ℚ_p))
    ≃⊕_r ≥ 0 R Γ_dR(X_K/K)/Fil^≥ r
    R Γ_dR(X_K/K) [2r].

 


In particular, 
<Ref> applies for i = 0 and overlaps with the results of
<cit.>, although it does not identify the
obstruction class with the crystalline Chern character. 
In this subsection, we observe that <Ref> can be extended to
essentially arbitrary formal schemes, using comparisons between cyclic and de
Rham cohomology as in <cit.>. 
This argument will not essentially rely on having a fiber square as in
Theorem [thm:a]A
(versus a fiber sequence), and could be deduced from the results of
<cit.>. 


    Let 𝔛 be a qcqs p-adic formal
	 scheme with bounded p-power torsion.
    Given i∈ℤ and a class x∈_i(_1;) there is a natural
 class
    
    c(x)∈⊕_r≥0 H^2r-i( 
    	L Ω_𝔛/ L Ω_𝔛^≥ r)_ℚ_p
 with the property that x lifts to
    _i^cts( 𝔛; ℚ) if and only if c(x)=0.
	 More precisely, there is a natural equivalence
	 of spectra
	
    cofib( ^cts(𝔛;
    	ℚ) →( 𝔛_1; ℚ) ) ≃⊕_r ≥ 0(
    	L Ω_𝔛/L Ω_𝔛^≥
    	r[2r])_ℚ_p .
 






It clearly suffices to exhibit the natural equivalence (<ref>), and for this we may assume
𝔛 = (R) is affine, since all terms satisfy Zariski descent. 
Now we have seen 
that the cofiber in (<ref>) 
can be identified with the 
cofiber of 
(R; ℚ_p) →(R/p; ℚ_p),
or equivalently with (R; ℚ_p)[2] by <Ref>. 
We invoke the result of <cit.> which constructs on
    (R;_p)[2]
    a natural exhaustive decreasing filtration
	 ^≥⋆(R;_p)[2] with graded pieces
    
    ^n(R;_p)[2] L Ω_R/ L Ω_R^≥ n[2n],

    where, as before,  L Ω_R is the p-adic derived de Rham cohomology of R and
    L Ω^≥⋆_R is the Hodge filtration on the derived de Rham cohomology
    (see specifically the proof
    of <cit.>).[The work
    of <cit.> was essentially motivated by that
    of <cit.> which among many other things established such
    filtrations for quasisyntomic rings by descent.]
   It follows that on (R;_p)[2] there is a natural exhaustive
    decreasing filtration Fil^≥⋆(R;_p)[2] with graded pieces
    
    ^n(R;_p)[2] ≃ (L Ω_R/L Ω_R^≥
    	 n)_ℚ_p[2n].

    An argument as in <cit.> can be
    used to show that there is an action of Adams operations on
    (R;_p)[2]
    where λ∈_p^× acts via λ^n on
    ^n(R;_p). In particular, these split (R;_p)[2] into
    eigenspaces so that there is a natural decomposition
    
    (R;_p)[2]⊕_n (L Ω_R/L Ω_R^≥
    	 n)_ℚ_p[2n].

    The result now follows from the Beilinson fiber sequence.


 
In the above work, ℤ_p was used as the base for cyclic and de Rham
cohomology, but often this is not essential. 
Suppose now that A is a commutative -algebra with
L_A/ quasi-isogenous to zero. Then it is not difficult to see that, for formal schemes over A, we can replace all occurrences of derived
de Rham cohomology relative to ℤ_p with such occurrences relative to
A. 





§ THE MOTIVIC FILTRATION ON TC


 
In this section (which will not use the Beilinson fiber square), we prove some
general structural results on topological cyclic homology TC (in
particular, Theorem G from the introduction) and on
the “motivic” filtration constructed by Bhatt–Morrow–Scholze <cit.>. 


Recall that, according to <cit.>, for R a quasisyntomic ring (see
<Ref> below for a review), 
TC(R; ℤ_p) admits a 
complete descending ℤ_≥ 0-indexed filtration Fil^≥⋆TC(R;ℤ_p) with associated graded terms given as
^i(R;_p)ℤ_p(i)(R)[2i]. In this section, we will prove some
structural properties of this filtration. 
Our main results are as follows. 




 


    
  (1)
Let R ∈ be a quasisyntomic ring. 
Then ℤ_p(i) (R) ∈ D^≤ i+1(ℤ_p). 
Consequently, we have Fil^≥ i(R; ℤ_p) ∈_≥
i-1. 

  (2)
The functors R ↦ℤ_p(i)(R) and R↦Fil^≥ i(R;
ℤ_p) are left Kan extended from finitely generated
p-complete polynomial ℤ_p-algebras. 

 

Part (2) was indicated to us by Scholze. 
In view of it, we can extend the construction of the ℤ_p(i) to all (p-complete) 
rings. 

 

Let (R, I) be a henselian pair where R and R/I are p-complete.
Then fib( ℤ_p(i)(R) →ℤ_p(i)(R/I)) ∈ D^≤
i(ℤ_p). 
 

In particular, using the known description in characteristic p, we obtain that
for any R there is a complete description of the top cohomology H^i+1(
𝔽_p(i)(R)) and that this vanishes étale locally.






 §.§ Review of <cit.>

Here we recall some of the major results and techniques of <cit.>. 
We recall first  the quasisyntomic site  
 (a non-noetherian version of the syntomic site used by
Fontaine–Messing <cit.>)
 and the subcategory ⊂ of quasiregular semiperfectoid rings. 

 
Let R be a commutative ring. An R-module M
is called p-completely flat (resp. p-completely faithfully
flat) if 
M ⊗^_R (R/p) ∈ D(R/p) is a flat (resp. faithfully flat)
R/p-module concentrated in degree zero. 
Similarly, an object N ∈ D(R) has p-complete
Tor-amplitude in [a,b] if N ⊗^_R R/p ∈ D(R/p) has
Tor-amplitude in [a, b]. 
 


 



  * A commutative ring R is called quasisyntomic
if it is p-complete, has bounded p-power torsion, and Ł_R/ℤ_p
 has p-complete Tor-amplitude in [-1, 0] (indexing conventions for the derived category are
cohomological).
We let  be the category of quasisyntomic rings, with all ring homomorphisms.

  * The category  (or more precisely its opposite) acquires the structure of a site as follows: a map A→
B in  is a cover
if A → B is p-completely faithfully flat and if Ł_B/A∈ D(B) has  p-complete Tor-amplitude in
[-1, 0]. 
We call a map with all of the above properties, except that A → B only
assumed p-completely flat (rather than p-completely faithfully flat), a quasisyntomic map. 

  * An object R ∈ is quasiregular semiperfectoid
if R admits a map from a perfectoid ring and the Frobenius 
on R/p is surjective. 
We let ⊂ be the full subcategory spanned by quasiregular
semiperfectoid rings. 
If R is additionally an 𝔽_p-algebra, then R is called
quasiregular semiperfect.

 



For future reference, we will also 
need the relative versions
qSyn_A and A of the quasisyntomic sites (of which the first is
considered in <cit.>). 

 
Fix a quasisyntomic ring A ∈. 
We define the sites qSyn_A and A as follows. 


  * We let A denote the category of
A-algebras
B which are quasisyntomic as underlying rings and such that L_B/A∈ D(B) has
p-complete Tor-amplitude in [-1, 0].  
We let qSyn_A ⊂A be the full subcategory spanned by
the quasisyntomic A-algebras (i.e., those B such that B is additionally
p-completely flat over A). 

  * We make A and qSyn_A into sites by declaring a cover to be a map which is a cover
in . 

  * We let A (resp. _A) denote the subcategory of A
(resp. qSyn_A) spanned by
A-algebras 
whose underlying ring is quasiregular semiperfectoid. 
Note that if B ∈A, then the p-completion of L_B/A[-1] is a
p-completely flat, discrete B-module by <cit.>.

 

Note that in the case A = ℤ_p, 
qSyn_ℤ_p is the category of p-torsion free quasisyntomic
rings and ℤ_p =. 
For A = 𝔽_p, 
𝔽_p and qSyn_𝔽_p are both simply the
subcategory of  spanned by those quasisyntomic rings which are
𝔽_p-algebras <cit.>; more generally A is
the category of A-algebras which are quasisyntomic 
for any perfectoid ring A. 

The site  has a basis given 
by , and similarly in the relative cases. All the functors below will be sheaves on ; to describe them,
it therefore suffices to describe them as sheaves on  <cit.>. 

We now review the prismatic sheaves  on , constructed via topological Hochschild and cyclic homology. A
purely algebraic construction via the prismatic cohomology of Bhatt–Scholze
is given in 
<cit.> (at least for algebras over a base perfectoid ring), which also produces objects before Nygaard completion. 


The objects _R{i} and 𝒩^≥ n_R define
sheaves on  with values in D(ℤ_p)^≥ 0. Each of these sheaves is constructed  via descent <cit.> from ⊂, on which they take discrete values defined via topological Hochschild
homology.



  * For R ∈, (R;
ℤ_p) is concentrated in even degrees, so the homotopy fixed
point and Tate spectral sequences for ^-(R; ℤ_p) and (R;
ℤ_p) 
degenerate and (R; ℤ_p) is 2-periodic.
For R ∈, we have 
    _R = π_0( ^-(R; ℤ_p)) = π_0( (R; ℤ_p)).
 

  * For R ∈ and n∈ℤ, the ideal 𝒩^≥ n_R ⊂_R is the one
defined by the homotopy fixed point spectral sequence, i.e.,
 
    𝒩^≥ n_R = im( π_0(
     (τ_≥ 2n(R;
    ℤ_p))^hS^1) →π_0 ((R; ℤ_p)^hS^1)
    ).


  * For i∈ℤ we further have the invertible -modules (as
sheaves on ) {i}, called Breuil–Kisin twists. For R ∈,

    _R{i} = π_2i ( (R; ℤ_p)),

and by 2-periodicity _R{i} =
_R{1}^⊗ i. 
We have a natural isomorphism 
 
    𝒩^≥ i_R {i}≃π_2i( ^-(R;ℤ_p)).

More generally,
𝒩^≥ n + i_R{i}
is the image of the injection π_2i((τ_≥ 2n + 2i(R;ℤ_p))^hS^1) →π_2i(((R;ℤ_p))^hS^1) for n∈ℤ. 

  * There are two maps 
of sheaves on ,  
    can, φ_i𝒩^≥ i_R
    {i}⇉_R {i}
 arising from the canonical and Frobenius maps
TC^-(R; ℤ_p) ⇉TP(R; ℤ_p);
in particular, we obtain an endomorphism φ = φ_0_R →_R.

  * Finally, the map ^-(R; ℤ_p) → R yields a projection map 
a_R_R → R, a surjection with kernel 𝒩^≥ 1_R
⊂_R. 







[Perfectoid rings, <cit.>] 
Let R_0 be a perfectoid ring. 
    In this case, we have Fontaine's ring (R_0)=W(R_0^♭) and the surjective map
θ(R_0) → R_0, whose kernel is a principal ideal generated by a
nonzerodivisor ξ∈(R_0); θ is the universal pro-nilpotent,
p-complete thickening of R_0. 
We have a canonical isomorphism _R_0≃(R_0)
such that the projection map a_R_R_0→ R_0 is θ. 
There are also non-canonical isomorphisms _R_0{i}≃(R_0) for each i. 
The map φ = φ_0_R_0→_R_0 is given by the
Witt vector Frobenius on (R_0). 
The Nygaard filtration on _R = (R_0) is the ξ-adic filtration. 
The map ϕ_i is injective (and ϕ_0-semilinear) and its image
is given by ( ξ^-min(i, 0)) for ξ  =
φ(ξ). 

 



For i ≥ 0, 
the sheaf ℤ_p(i) is defined as the homotopy equalizer of 
, ϕ_i, i.e., via 

    _p(i)fib(^≥ i_R{i}_R{i}).

Consequently, since TC is itself a homotopy equalizer,
we also have
for R ∈,

    ℤ_p(i)(R) = (τ_[2i-1, 2i]TC(R;
    ℤ_p))[-2i].

We also define quasisyntomic sheaves 𝔽_p(i) and
ℚ_p(i) by reducing ℤ_p(i) modulo p or inverting p on
_p(i),
respectively. 


Using the result <cit.> that TC defines a sheaf for the fpqc topology on
rings, one sheafifies the Postnikov filtration and obtains the following fundamental result.

 

Let R ∈. Then 
(R; ℤ_p), ^-(R; ℤ_p), (R; ℤ_p), and (R;
ℤ_p)
naturally upgrade to filtered spectra  with complete,
multiplicative descending filtrations
^≥⋆(R; ℤ_p), ^≥⋆^-(R;
ℤ_p), ^≥⋆(R; ℤ_p), and
^≥⋆(R; ℤ_p), indexed by ℤ_≥0, ℤ, ℤ, and ℤ_≥0 respectively,
such that the associated graded pieces are given by

    
  (1) 
^i (R; ℤ_p) ≃𝒩^i _R{i}[2i]
def=( 𝒩^≥ i_R{i}/𝒩^≥ i+1_R{i}) [2i] for all i≥0; moreover, in this case the
Breuil–Kisin twists can be trivialized, so 
also ^i (R; ℤ_p)≃𝒩^i _R[2i],

  (2) ^i ^-(R; ℤ_p) = 𝒩^≥ i_R{i} [2i] for all i ∈ℤ,

  (3)  ^i (R; ℤ_p) = _R{i}[2i] for all i∈ℤ,

  (4)  ^i (R; ℤ_p) = ℤ_p(i)(R)[2i] for all
i ≥ 0.


 

 
Recall that for p-adic rings, TC and p-adic étale -theory
agree in nonnegative degrees. Cf. <cit.> for smooth algebras in characteristic p, and in general <cit.>. 
One may thus expect 
the filtration of 
<Ref> to be the étale sheafification of the filtration on algebraic
-theory 
 with associated graded motivic cohomology,
cf. <cit.> (for smooth schemes over fields). 
In particular, one expects the ℤ_p(i) to be some form 
of p-adic étale motivic cohomology. This is essentially understood in equal
characteristic (already by <cit.>), as we review below, but has not yet
appeared in mixed characteristic. In mixed characteristic and under
finiteness assumptions (e.g., smooth schemes over a DVR), many
authors have studied étale motivic cohomology <cit.> and similar “p-adic étale Tate twists,” e.g.,
those of 
<cit.>, though the construction is very different from that
of <cit.>; one ultimately hopes to compare all of them, and we will at least
offer some information in this and the next section.  
 

We review the discreteness property of the ℤ_p(i). By construction, 
the objects 𝒩^≥ i_R{i} are sheaves on
 with values in D(ℤ_p)^≥ 0 (recall <cit.> that such sheaves form the coconnective
part of the derived ∞-category of the category of abelian sheaves on );
as objects of this category, they are in fact
discrete, since they take discrete values on the basis .  
A deep result of Bhatt–Scholze (conjectured in <cit.> and proved in the
characteristic p case there) is that this
discreteness also holds for the ℤ_p(i),
although they in general take values in cohomological degrees [0, 1] for rings in .


 

The D(ℤ_p)^≥ 0-valued sheaf ℤ_p(i) on 
is discrete and torsion free. More precisely, given R ∈, there is a
cover R → R' in  such that ℤ_p(i)(R') is discrete and
torsion free. 
 




Finally, we review the prism
structure on _R, for R quasiregular semiperfectoid. For simplicity, we
will assume R to be p-torsion free.

 

Let R ∈_ℤ_p. Suppose R is an algebra over the perfectoid
ring R_0, with notation as in 
<Ref>. 
Then (R; ℤ_p)/ξ≃(R; ℤ_p)^tC_p, and this is concentrated in even
degrees and p-torsion free. 
 
 
We have that 

    (R; ℤ_p)/ξ≃(R / R_0 ; ℤ_p)
 by <cit.>. 
Since R is p-torsion free and quasiregular semiperfectoid, we find that
(R/R_0; ℤ_p) is concentrated in even degrees and is p-torsion free, where it
is given by Hodge-complete p-adic derived de Rham cohomology by
<cit.>. 
In particular, it follows that (ξ, p) defines a regular sequence
on _R. 
Since _R is complete with respect to this ideal, it follows that (p,
ξ) is a regular sequence; since ξ^p ≡ξ
(mod p), we get that (p, ξ) is a regular sequence,
and hence so is (ξ, p). 
Now, the equivalence 
(R; ℤ_p)/ξ≃(R; ℤ_p)^tC_p is <cit.>, from which the
remainder now follows. 
 

[The prismatic structure on _R, cf. <cit.>] 
Let R ∈_ℤ_p. Suppose R is 
an algebra over the perfectoid ring R_0. 
Then the  ring _R = π_0( (R; ℤ_p)) has the structure 
of a prism (in the sense of <cit.>).  




  * We have the endomorphism φ  = φ_0, which is
congruent to the Frobenius modulo p, by <cit.>, and thus defines a δ-structure on
_R by p-torsionfreeness. 


  * We have the ideal I ⊂_R given by I = (ξ); I is the kernel of
π_0( (R; ℤ_p)) →π_0 ( (R; ℤ_p)^tC_p) and
therefore does not depend on the choice of R_0. 


 Finally, there is a natural map 

    η_R  R →_R/I,
 
given via the cyclotomic Frobenius (R; ℤ_p) →(R;
ℤ_p)^tC_p = (R; ℤ_p)/ξ upon applying π_0. 
 
 
In fact, by <cit.>, _R is the Nygaard completion of
the absolute 
prismatic cohomology of R, although we will not need this fact. 
 




 §.§ Relative  and its filtration


In this subsection and the next, we will prove connectivity bounds for the 
motivic filtration on . 
We will prove that for any R ∈, we have ^≥ n(R;
ℤ_p) ∈_≥ n and  
𝒩^n_R ∈ D^≤ n(ℤ).
It is not difficult to deduce the above connectivity bound in the case R is
an algebra over a fixed perfectoid ring, using methods as in <cit.>; see in particular <cit.>. To verify the connectivity bound in the general case, we will use additionally a
fiber sequence which arises from the work of 
Krause–Nikolaus <cit.>, which gives a comparison between relative and
absolute . 

Let 𝒪_K denote a complete discrete valuation ring of mixed characteristic
(0, p) with perfect residue field k; let π∈𝒪_K be a
uniformizer. 
The primary case of interest is 𝒪_K = ℤ_p and π = p. 

[Relative topological Hochschild homology] 
Let R ∈𝒪_K. 
We consider the 𝔼_∞-ring 𝕊[z] and consider R as an
𝕊[z]-algebra
via z ↦π. Using this, we can form 
the relative topological Hochschild homology (with p-adic coefficients)
(R/𝕊[z]; ℤ_p). 
The construction R ↦(R/𝕊[z]; ℤ_p) defines a sheaf of spectra on
𝒪_K, thanks to <cit.> (which gives that 
R ↦(R/𝒪_K; ℤ_p) is a sheaf)
and
(<ref>) below. 
We observe the following two comparisons for relative .


  * When we base-change along the map 𝕊[z] →𝕊 where z↦ 0, we find that 
 
    (R/𝕊[z]; ℤ_p) ⊗_𝒪_K k ≃(R ⊗^L_𝒪_K k;
    ℤ_p).
  

  * We have an equivalence
 
    (R/𝕊[z]; ℤ_p)
         ⊗_( 𝒪_K/𝕊[z]; ℤ_p)𝒪_K ≃(R/𝒪_K; ℤ_p)  .
 
Thus, 
(R/𝕊[z]; ℤ_p) is a deformation of Hochschild homology relative
to 𝒪_K. 


 


Next, we need an analog of  the
Hochschild–Kostant–Rosenberg theorem for relative  (in the absolute case
for algebras over a perfectoid ring,
this is <cit.> and <cit.>). 

 
Let R be a formally smooth 𝒪_K-algebra. 
Then we have a natural isomorphism of graded rings

    _*(R/𝕊[z];ℤ_p) ≃Ω^∗_R/𝒪_K
    [σ] ,   |σ| = 2,

    where Ω^∗_R/𝒪_K denotes the
    p-completion of the de Rham complex of R over _K.
 
 
In the case R = 𝒪_K, this follows from Bökstedt's
calculation of (𝔽_p), cf. <cit.>,
<cit.>,  or <cit.>. 
Now (<ref>) shows that 
(R/𝕊[z]; ℤ_p)/σ≃(R/𝒪_K;
ℤ_p), and the Hochschild–Kostant–Rosenberg theorem yields _*(R/𝒪_K;
ℤ_p) ≃Ω^∗_R/𝒪_K. 
It remains to show that the induced Bockstein spectral sequence  for 
(R/𝕊[z]; ℤ_p)
(with respect to
taking the cofiber of σ) degenerates, or equivalently that the map of the
HKR isomorphism lifts to a map Ω_R/𝒪_K^∗→_*(R/𝕊[z]; ℤ_p).  
Indeed, since (R/𝕊[z]; ℤ_p) is an 𝔼_∞-algebra with an
S^1-action receiving a map from (𝒪_K/𝕊[z]; ℤ_p), we obtain the structure of a commutative differential graded
algebra on 
_*(R/𝕊[z]; ℤ_p), and it receives a map (of cdgas) from
𝒪_K[σ] with trivial differential. The universal property of the
de Rham complex now produces the desired map 
Ω^∗_R/𝒪_K→(R/𝕊[z]; ℤ_p). 
 

Left Kan extending from finitely generated p-complete polynomial 𝒪_K-algebras, we obtain the following result, which is proved exactly as in
<cit.>; the key point is that any R ∈𝒪_K
has the property that the p-completion of L_R/𝒪_K is the
shift of a p-completely flat R-module.

 

If R ∈𝒪_K,
then (R/𝕊[z]; ℤ_p) is concentrated in even degrees, and
each π_2n(R/𝕊[z]; ℤ_p) is a p-completely flat R-module. 
Furthermore, the R-module
π_2n(R/𝕊[z]; ℤ_p)
admits a natural finite increasing filtration with graded pieces
the (discrete and p-completely flat) R-modules
(⋀^j L_R/𝒪_K) [-j] for j ≤ n,
where ⋀^j L_R/𝒪_K denotes the p-completion
    of ⋀^j L_R/𝒪_K.
 

[The filtration on relative ] 
Let R ∈𝒪_K. 
In <cit.>, a multiplicative, convergent ℤ_≥ 0-indexed filtration 
^≥⋆(R/𝕊[z]; ℤ_p)
on (R/𝕊[z]; ℤ_p)
in sheaves of spectra on 𝒪_K is defined.[Actually,
in loc. cit, the filtration is defined only on those objects which are flat
over 𝒪_K, but the arguments do not require this.] This filtration 
is defined such that it restricts to the double speed Postnikov filtration for R ∈𝒪_K, i.e., ^≥ n(R/𝕊[z]; ℤ_p) =
τ_≥ 2n(R/𝕊[z]; ℤ_p) for such R. 
By <Ref> and <cit.>, the associated graded
    pieces of the Postnikov filtration on 
(-/𝕊[z]; ℤ_p) on 𝒪_K are
sheaves; thus, one unfolds and  obtains the filtration for all 
R ∈𝒪_K. 
 


 

Let R ∈𝒪_K.
Then gr^n (R/𝕊[z]; ℤ_p) admits a natural finite
increasing filtration
with associated graded 
( ⋀^j L_R/𝒪_K)[2n-j] for j ≤ n. 
In particular, we find that gr^n(R/𝕊[z]; ℤ_p) ∈_≥ n and ^≥ n(R/𝕊[z]; ℤ_p) ∈_≥ n. 
Furthermore, the constructions 
    R ↦gr^n (R/𝕊[z];
        ℤ_p)  and  
    R↦^≥ n(R/𝕊[z]; ℤ_p)
 (as functors on
𝒪_K to
p-complete spectra) are left Kan extended from
finitely generated  p-complete polynomial 𝒪_K-algebras. 

 
 
The first assertion follows from <Ref> by unfolding; the
connectivity assertions then follow in turn. 
Since the cotangent complex and its wedge powers are left Kan extended from
finitely generated polynomial algebras, the last assertion follows too. 
 




 §.§ Preliminary connectivity bounds


We use the spectral sequence of Krause–Nikolaus <cit.> to obtain a
relationship between the 
relative and absolute . 

 

If R ∈𝒪_K,
then there exist natural surjective maps 
f_n π_2n(R/𝕊[z];ℤ_p) →π_2n-2(R/𝕊[z];ℤ_p)
and natural isomorphisms 
π_2n(R; ℤ_p) ≃ker(f_n). 
 
 
Recall that both (R; ℤ_p) and (R/𝕊[z]; ℤ_p) are
concentrated in even degrees since R ∈𝒪_K (see
<cit.> and <Ref>). 
Therefore, the result follows directly from <cit.>; the spectral
sequence of loc. cit. must degenerate after the first differential, and the
maps of the first differential must be surjective or one would have odd degree
contributions to (R; ℤ_p). 
 

The following fiber sequence (<ref>) will be the basic tool in obtaining connectivity bounds
on the filtration on  and its variants.

 

If R ∈𝒪_K, then for each n there is a natural fiber sequence

    gr^n (R; ℤ_p) →gr^n(R/𝕊[z]; ℤ_p) →gr^n-1(R/𝕊[z];
    ℤ_p)[2].

In particular, we have gr^n (R; ℤ_p),
Fil^≥ n(R; ℤ_p) 
∈_≥ n for any R ∈𝒪_K.  
Finally, the functors R ↦gr^n (R; ℤ_p) and
R↦Fil^≥ n(R; ℤ_p) on  are left Kan extended 
from finitely generated p-complete polynomial ℤ_p-algebras, as functors to p-complete spectra. 
 

 
The fiber sequence follows from 
<Ref> by unfolding in R. 
The connectivity assertion 
for gr^n (R; ℤ_p)
then follows from <Ref>; the assertion for Fil^≥
nTHH(R; ℤ_p) then follows since the filtration is complete. 
The Kan extension assertion for Fil^≥ n(R;
ℤ_p) also follows from 
the one for Fil^≥ n(R/𝕊[z]; ℤ_p)
as in 
<Ref> (taking 𝒪_K = ℤ_p). 
 

 


    
  (1) 
If R ∈,
then 𝒩^n _R ∈ D^≤ n(ℤ_p). 

  (2)
If R → R' is a surjective map in , then 
fib( 𝒩^n _R →𝒩^n _R') ∈ D^≤
n(ℤ_p). 

 
 
Part (1) is a special case of <Ref> (take 𝒪_K = ℤ_p and π = p). 

For part (2), note that the hypothesis implies that R→ R' induces a surjection on H^0 of p-completed cotangent complexes over ℤ_p, and similarly on any wedge power. It then follows from <Ref> that
fib( ^n (R/𝕊[z]; ℤ_p)[-2n] →^n (R'/𝕊[z]; ℤ_p)[-2n]) ∈ D^≤ n(ℤ_p), whence we conclude by (<ref>).
 

Our main general connectivity bound is <Ref> below. 
To formulate it, we need to be able to twist the ideal 
I ⊂_R from <Ref>.
First, we observe that this ideal is also trivialized after base-change along
a_R. 

 
Let R ∈_ℤ_p, and let I ⊂_R denote the ideal
defining the prism structure. 
Then there is a natural isomorphism I ⊗__R R ≃ R, i.e.,
the ideal is naturally trivialized after base-change along a_R_R → R. 
 
 
Observe that the base-change I ⊗__R R defines a functorial
choice of invertible R-module, for any R ∈_ℤ_p. 
By faithfully flat descent, we obtain for any R ∈qSyn_ℤ_p a 
choice of invertible R-module, which is functorial in R. 
Choosing a trivialization over R = ℤ_p, we obtain a functorial
trivialization everywhere. 
 

 
For s, i , n ≥ 0, 
we let R ↦ I^s 𝒩^≥ n_R {i} denote
the D(ℤ_p)^≥ 0-valued sheaf on 
qSyn_ℤ_p
defined by unfolding 
the discrete sheaf on _ℤ_p defined by the aforementioned formula, for
I ⊂_R the ideal defining the prismatic structure. 
For R ∈_ℤ_p, 
since I defines a Cartier divisor in _R, we have
I^s 𝒩^≥ n_R {i}≃ 
I^s ⊗__R 𝒩^≥ n_R {i}. 
 

 

Let R ∈qSyn_ℤ_p and i, n, s ≥ 0. Then the cofiber
I^s_R{i}/ I^s 𝒩^≥ n_R {i}
belongs to D^≤ n-1( ℤ_p). 
Moreover, this cofiber is left Kan extended from finitely generated
p-complete polynomial
ℤ_p-algebras. 
 

By dévissage it suffices to show that 
    I^s𝒩^n_R{i}∈ D^≤ n(R) for each n≥0 and
that this is left Kan extended from finitely generated p-complete polynomial ℤ_p-algebras. Here we write I^s𝒩^n_R{i} for the unfolding from _ℤ_p of I^s ⊗__R 𝒩^n_R {i}.
However, the twists here are trivialized by <Ref> since
𝒩^n _R is an R-module, 
so that
I^s𝒩^n_R{i}≃𝒩^n_R.
Thus, the result follows from <Ref> as well as
<Ref> (for the left Kan extension assertion). 
 


We finish this subsection by recording a connectivity bound that depends on the number of generators of the cotangent complex (we will not use this result in the paper, but note that it implies in particular that _𝒪_K∈ D^≤ 1(ℤ_p)).

 
Let R be a commutative ring, and let M ∈ D^≤ 0(R). 
Suppose H^0(M) is generated by d elements. 
Then for all j, (⋀^j M) [-j] ∈ D^≤ d(R). 

 

 
The result is clear if M = R^d itself.
In general, we have a map R^d → M inducing a surjection on H^0, so the cofiber F of
the map satisfies F ∈ D(R)^≤ -1. 
It follows that ⋀^j' F [-j'] ∈ D(R)^≤ 0 for all j' by
standard connectivity estimates (see <cit.> for an account).  
Using the natural filtration on ⋀^j  M [-j] with associated graded
terms
⋀^j' F[-j'] ⊗_R ⋀^j-j' R^d [-(j-j')], 
the result easily follows. 
 

 

Let R  ∈𝒪_K, let n,i≥0, and suppose that H^0( L_R/𝒪_K) is generated by d elements. Then 𝒩^n _R and  𝒩^≥ n_R {i} lie in D^≤ d+1(R). 


By <Ref>,  ^n (R/𝕊[z]; ℤ_p)
has a finite filtration with graded pieces
⋀^j L_R/𝒪_K [2n-j] for 0 ≤ j ≤ n. 
By <Ref>, we find that 
^n (R/𝕊[z]; ℤ_p) ∈ D^≤ d-2n(R). 
Using the fiber sequence (<ref>), we find now that ^n (R;
ℤ_p) ∈ D^≤ d - 2n+1(R). Shifting by 2n the result now
follows for 𝒩^n_R. The same connectivity bound then follows for each 𝒩^≥ n_R {i}/𝒩^≥ n+r_R {i} by dévissage, and then for 𝒩^≥ n_R {i} by passing to the limit.
 





 §.§ Frobenius nilpotence on _R/p, and proof of <Ref>
(2)

In this subsection, we record some results about the contracting property of Frobenius on
_R/p and use it to prove part of <Ref>. 
If R ∈_ℤ_p is a p-torsion free quasiregular
semiperfectoid ring, then both _R and all graded steps 𝒩^n_R of the Nygaard filtration are p-torsion free (e.g., because _*(R; ℤ_p) is p-torsion free and concentrated in even degrees). 
 For i, r ≥ 0, we will consider the maps 

    , ϕ_i𝒩^≥ i+r_R{i}/p  →_R{i}/p .

and show that both maps respect the I-adic filtration from <Ref>, with
ϕ_i inducing the zero map on associated graded pieces in positive degrees
(<Ref>). We will show in addition that
can - ϕ_i induces an automorphism of 𝒩^≥ i + r_R{i}/p for r ≫ 0 (<Ref>). 

 

Let R ∈_ℤ_p and i,r≥0. Then
the map 
ϕ_i𝒩^≥ i_R{i}→_R{i}
 carries 
𝒩^≥ i + r_R{i} into 
I^r_R{i}. 
 
 
Let R_0 be a perfectoid ring mapping to R and fix ξ, ξ∈(R_0) as usual. 
Then <cit.> we have an isomorphism
^-_*(R_0; ℤ_p) ≃(R_0)[u, v]/(uv - ξ) for |u| = 2, |v|
= -2. 
In this case, the filtration on 𝒩^≥ i_R{n}≃π_2i( ^-(R; ℤ_p))
is the filtration by powers of v:

    𝒩^≥ i+r_R {i} = v^r  π_2i+2r^-(R;
    ℤ_p) ⊂π_2i^-(R; ℤ_p).

But (as in loc. cit.) the cyclotomic Frobenius carries v to a multiple of ϕ(ξ) =
ξ in
π_-2(R_0; ℤ_p); recalling that I=(ξ̃), the result follows.  
 

[The I-adic filtrations modulo p] 

Let R ∈_ℤ_p. For each i,s,r≥0, the map ϕ_i𝒩^≥ i+r_R{i}/p→_R{i}/p is Frobenius semi-linear by <Ref>(1), and so carries I^s(𝒩^≥ i+r_R {i}/p) to I^ps + r(_R{i}/p) by
 <Ref>. But we have seen in <Ref> that
 (p,ξ̃) and (ξ̃,p) are regular sequences on _R, whence
 the canonical maps are isomorphisms I⊗__R_R/p≃
 I(_R/p)≃ I/p, and similarly for any power of I and Breuil–Kisin
 twist of _R.  We thus get maps 

    , ϕ_i I^s ⊗__R𝒩^≥ i+r_R
    {i}/p →
    I^s ⊗__R_R {i}/p.

 

For convenience, we record what we have proved about the interaction of the
Frobenius and the I-adic filtration, as it will be used to prove 
<Ref>:


 

Let R ∈_ℤ_p, and let i,r≥0. 
The maps 
(<ref>) upgrade to the structure of filtered maps with respect to the
I-adic filtrations on both sides, i.e., there are compatible 
maps for each s ≥ 0, 

    , ϕ_i I^s ⊗__R𝒩^≥ i+r_R{i}/p → 
    I^s ⊗__R_R{i}/p.
 
Furthermore, the map  ϕ_i induces the zero 
map on associated graded pieces unless s = r = 0. 
 
 
In <Ref> we constructed the maps and showed that in fact ϕ_i  has image in I^ps+r⊗__R_R{i}/p.
 

For the moment we need the following consequence of our arguments.



Let R ∈_ℤ_p and i≥0. For r≫0 (independent of R), the map
- ϕ_i𝒩^≥ i_R{i}/p →_R{i}/p 
 carries 𝒩^≥ i+r_R{i}/p
 isomorphically onto itself. Consequently, for such r, one has a natural isomorphism
 
    𝔽_p(i)(R) ≃fib(  - ϕ_i (𝒩^≥ i_R{i}/
    𝒩^≥ i+r_R {i})/p → 
    ( _R{i}/
    𝒩^≥ i+r_R {i}) /p)
     .

 
 
This is <cit.>. 
By <Ref> above (and choosing as usual a
perfectoid ring R_0 mapping to R), 
we find that φ_i carries 𝒩^≥ i +
r _R {i}/p 
(equivalently, v^r 
𝒩^≥ i+r_R{i+r}/p)
into multiples of ξ^r_R
{i}/p. Since we are working modulo p, we have 
ξ^r_R
{i}/p = 
ξ^rp_R{i}/p
 ⊂𝒩^≥ rp_R{i}/p.  For r ≫ 0, this
 is contained in 𝒩^≥ i  + r + 1_R{i}/p, whence we have shown that 
ϕ_i carries 𝒩^≥ i+r_R{i}/p to 
𝒩^≥ i+r+1_R{i}/p. 

It follows that - ϕ_i carries 
𝒩^≥ i+r_R{i}/p into itself, and it differs from the identity by a topologically nilpotent endomorphism
of 
𝒩^≥ i+r_R{i}/p with respect to the Nygaard
filtration. Therefore it is an isomorphism and the result follows.
 

 

Let F, G→ D(ℤ_p) be p-complete quasisyntomic sheaves 
equipped with complete descending ℤ_≥ 0-indexed filtrations Fil^≥⋆ F and
Fil^≥⋆ G. 
Let F → G be  a map of functors (not necessarily filtration-preserving). If

    
  (1) for R ∈_ℤ_p, the objects
^r F(R) and ^r G(R) are discrete, p-complete, and p-torsion free (and therefore so
are F(R), G(R)),

  (2)
each of the associated graded terms gr^r F and gr^r G is
 left
Kan extended from finitely generated p-complete polynomial ℤ_p-algebras to the
p-complete derived category, and

  (3) 
there exists N such that for r ≥ N and 
for R ∈_ℤ_p, the map F(R)/p → G(R)/p 
carries the submodule Fil^≥ r F(R)/p isomorphically to Fil^≥
r G(R)/p (so that Fil^≥ NF(R)/p →Fil^≥
NG(R)/p is an equivalence of filtered
abelian groups),

then fib( F → G)  is p-completely left Kan extended
from finitely generated p-complete polynomial ℤ_p-algebras. 
 
 
It suffices to check that 
fib( F → G)/p is left Kan extended from finitely generated
p-complete polynomial algebras.
Let F', G' denote the functors on  obtained by restricting F, G to
finitely generated p-complete polynomial algebras and then left Kan extending to
, with their left Kan extended filtrations.  
Our assumptions imply that F, G are the respective completions of F', G' with respect
to their filtrations. 

It suffices to check that 
the natural map induces an equivalence fib(F' → G')/p ≃fib(F → G)/p. 
We claim that for any R ∈, there are natural commutative diagrams,
compatible in r ≥ N,
    
    Fil^≥ r F'(R)/p [d]  [r]^≃   Fil^≥ r
    G'(R)/p [d]  
    
    F'(R)/p [r]      G'(R)/p.

In fact, it suffices to prove this by left Kan extension for R
finitely generated p-complete polynomial
over ℤ_p, and then  we can replace F', G' by F, G. By descent for
F, G,
we can then reduce to R ∈_ℤ_p,  whence we have the desired
diagrams by hypothesis. 

Using the diagrams (<ref>), we find that 
there is a natural commutative diagram

    F'(R)/p [d]  [r]      G'(R)/p [d]  
    
    F(R)/p [r]      G(R)/p

which is homotopy cartesian (taking the inverse limit over r). 
We finally find that fib(F → G) = fib(F' → G'), which
is left Kan extended from finitely generated p-complete polynomial algebras as desired. 
 


We show that  R ↦ℤ_p(i)(R), as a functor on , is left Kan extended from
finitely generated 
p-complete polynomial ℤ_p-algebras. 
Since ℤ_p(i)(R) = fib(   - ϕ_i𝒩^≥ i_R{i}→_R{i}), we will apply
    <Ref>
with F = 𝒩^≥ i{i}, G =
{i}
using the Nygaard filtrations
R ↦𝒩^≥ i+r_R{i} on .
Indeed, the associated graded terms for the Nygaard filtration are left Kan extended from
finitely generated p-complete polynomial algebras (<Ref>), and they are
torsion free on R ∈_ℤ_p. 
The last hypothesis follows from 
<Ref>. 
Then <Ref> gives that the ℤ_p(i) are left Kan extended from
finitely generated p-complete polynomial algebras as desired. 
Since R ↦(R; ℤ_p) is left Kan extended from finitely
generated p-complete polynomial rings (by <cit.> and since
TC(-; ℤ_p) commutes with geometric realizations on
simplicial commutative rings), it follows inductively
that the constructions R ↦^≥ i(R; ℤ_p) are also left
Kan extended from finitely generated p-complete ℤ_p-algebras. 


For future reference, we observe that we can obtain a motivic filtration on
TC for any simplicial commutative ring, by left Kan extension. 
We let SCR denote the ∞-category of simplicial commutative
rings. 

[Left Kan extending to SCR]

We have seen that the functor which sends R ∈ to the filtration
Fil^≥⋆(R; ℤ_p) is left Kan extended
from finitely generated p-complete polynomial algebras. 
Thus, we can left Kan extend to all p-complete simplicial commutative rings to obtain a functor
R ↦Fil^≥⋆(R; ℤ_p),
from SCR to p-complete filtered spectra, which commutes with
sifted colimits. We define functors ℤ_p(i) on SCR as
ℤ_p(i)(R) = gr^i TC(R; ℤ_p)[-2i], or in other words by left Kan extending ℤ_p(i) from finitely generated p-complete polynomial algebras.

Once we complete the proof of <Ref> in the next subsection, it will
follow that Fil^≥ iTC(R; ℤ_p) (resp. ℤ_p(i)) belongs to _≥
i-1 (resp. D^≤ i+1(ℤ_p)) for each i, by left Kan extending the connectivity estimate from the quasisyntomic case.

We also emphasize that the above proof of <Ref>(2) shows the following. Given i≥ 0, there is r≫0 such that for all p-complete rings R there is a natural expression  
    𝔽_p(i)(R) ≃fib(  - ϕ_i (𝒩^≥ i_R{i}/
    𝒩^≥ i+r_R {i}) ⊗_ℤ^L
    𝔽_p → 
    ( _R{i}/
    𝒩^≥ i+r_R {i}) ⊗_ℤ^L
    𝔽_p),

where the two Nygaard quotients on the right side are defined by left Kan extension from finitely generated p-complete polynomial algebras.
 





 §.§ Proofs of the connectivity bounds (<Ref>(1)) for the ℤ_p(i)

In this subsection, we complete the proof of 
<Ref>. 



Let can, ϕFil^≥⋆ M→Fil^≥⋆ N be maps of filtered objects in
D(ℤ) (with underlying objects M, N). Suppose that

    
  (1)  both filtrations are complete,
    
  (2) ϕ induces the zero map on associated graded pieces, and
    
  (3) there is a fixed r such that, for each s,
the induced map canFil^≥ s M →Fil^≥ s N has fiber in
D(ℤ)^≤ r.

Then can - ϕ M → N has fiber in D(ℤ)^≤ r.  
 
The fiber fib(  - ϕ M → N) acquires the natural structure
of a filtered spectrum, since , ϕ are filtered maps. 
On graded pieces, we find 
gr^s fib(  - ϕ M → N) ≃gr^s
fib(  M → N) since ϕ vanishes on associated graded
terms. In
particular, the associated graded terms belong to D(ℤ)^≤ r. 
Since the filtration on fib(  - ϕ) is complete, the
connectivity assertion on the fiber now follows from the analogous assertion on
associated graded terms. 
 




Let R ∈qSyn_ℤ_p. Then ℤ_p(i)(R) ∈ D^≤ i+1(
ℤ_p) for each i ≥ 0. 
 
 
First, we recall from the proof of <Ref> that 
the inclusion 𝒩^≥ i+1_R{i}→𝒩^≥
i_R {i} has cofiber in D^≤ i( ℤ_p). 
Using the resulting cofiber sequence, it thus suffices to show that the fiber of 
can - ϕ_i𝒩^≥ i+1_R{i}→_R{i}
belongs to D^≤ i+1( ℤ_p). 
Since everything is p-complete, it suffices to check this with mod p
coefficients. 


We consider the two maps

    can, ϕ_i𝒩^≥ i+1_R{i}/p →_R{i}/p.

Unfolding <Ref> shows that these upgrade to maps can, ϕ_i I^s𝒩^≥ i+1_R{i}/p →
I^s_R{i}/p for all s≥0, i.e., of I-adically filtered
    objects, and that the map ϕ_i acts trivially on associated graded
    pieces.

Furthermore, 
for each s≥0,
    the fiber of can I^s 
𝒩^≥ i+1_R{i}/p →
    I^s _R{i}/p
	 belongs to D^≤ i+1( ℤ_p) by
<Ref>.  
<Ref> (whose hypothesis (1) is satisfied by ξ̃-adic completeness in the case of R∈_ℤ_p) now shows that 
can - ϕ_i𝒩^≥ i+1_R{i}→_R{i}
belongs to D^≤ i+1( ℤ_p), as desired.
 



We wish to show that ℤ_p(i) ∈ D^≤
i+1(ℤ_p). 
But we have already proved part (2) of <Ref>, so the problem reduces to the case of finitely generated p-completely polynomial
rings over ℤ_p, which is covered by <Ref>. 
 






 §.§ Rigidity, and proof of <Ref>


In this subsection, we give a proof of <Ref>. Our strategy is to first
prove a continuity statement, after which Néron–Popescu and left Kan
extension arguments reduce the general case to that of a square-zero extension. In that case we use the automatic gradings that exist and argue with the pro-nilpotence of
Frobenius.
Recall that a ring R is said to be F-finite if R/p is finitely generated over its subring of p^th-powers.
The next result is an analog on graded pieces of 
<cit.>. 


Let R be a noetherian, F-finite, p-complete ring, and I⊆ R an ideal such that R is I-adically complete. Then the natural map ℤ_p(i)(R)→_s ℤ_p(i)(R/I^s) is an equivalence for any i≥0.


From <Ref> and completeness of the Nygaard filtration, it is enough
to prove the analogous continuity for each 𝒩^n{i}≃gr^n (-; ℤ_p)[-2n]. Then <Ref> reduces us further to continuity for each gr^n(-/𝕊[z]; ℤ_p), and finally the filtration of <Ref> reduces the problem to continuity for each ⋀^n L_-/ℤ_p.

A cell attachment lemma of André and Quillen (see
<cit.> for a presentation in this context)
shows that all cohomology groups of all wedge powers of L_(R/I^s)/R are pro
zero in s (except H^0 of ⋀^0 L). So the transitivity fiber
sequence 
shows that ⋀^nL_R/ℤ_p⊗^L_RR/I^s→⋀^nL_(R/I^s)/ℤ_p is a pro isomorphism on
all cohomology groups. This reduces the problem to showing that
⋀^nL_R/ℤ_p→_s⋀^nL_R/ℤ_p⊗_R^LR/I^s is an equivalence after p-adic completion. Since R has
bounded p-power torsion, this derived p-adic completion may be equivalently
computed as _r (-⊗^L_R R/p^r). Exchanging the limits, it is
enough to show that M≃_sM⊗^L_RR/I^s, where
M=⋀^nL_R/ℤ_p⊗^L_RR/p^r for any n≥0, r≥1. But
this follows from the facts that R is noetherian, that M is bounded above
(cohomologically), and that its cohomology groups are finitely generated
R-modules, cf. <cit.> (which uses F-finiteness).



As usual, one can prove stronger continuity statements when I=(p). For example,
given a p-complete ring R with bounded p-power torsion, we claim that
ℤ_p(i)(R)/p^r→{ℤ_p(i)(R/p^s)/p^r}_s induces an isomorphism of pro groups on all cohomology groups for any
fixed r≥1. To prove this we reduce to the case r=1 and appeal to
(<ref>) (instead of the completeness of the Nygaard filtration
used at beginning of the proof of <Ref>) to again reduce to the
analogous assertion for graded pieces of the Nygaard filtration. Then argue as
in the proof of <Ref> (the assumption that R has bounded
p-power torsion implies that the ideal (p) is pro Tor-unital in the sense of
<cit.>, which is needed to verify pro vanishing of
⋀^nL_(R/p^s)/R) to reduce to the analogous statement about
(⋀^nL_R/ℤ_p)/p→{(⋀^nL_R/ℤ_p)/p⊗_R^L R/p^s}_s, which follows from boundedness of the p-power torsion in R.



<Ref> holds in the  special case of henselian pairs of the form R=A
⊕ N, I=N, where A is the p-completion of a finitely generated
ℤ_p-polynomial algebra, N is a finitely generated A-module, and
R is the trivial square-zero extension of A by N.  


The proof of 
<Ref> will be given below. 
Using 
<Ref>, we explain how to deduce 
<Ref>. 


First, note that it is equivalent to prove that the homotopy fiber of <Ref> mod p belongs to D^≤ i(ℤ_p), i.e., we may replace ℤ_p(i) by 𝔽_p(i)=ℤ_p(i)/p.


We consider the functor on 
ℤ-algebras, 
R ↦ F(R) def=𝔽_p(i)( R_p) (where R̂_p denotes the
derived p-completion of R). By <Ref>, the functor F commutes with filtered colimits
in R. 
It suffices to show that if (R, I) is a henselian pair, then 
    fib(
    F(R) → F(R/I)) ∈ D^≤ i(𝔽_p) .



First, we prove (<ref>) in case I ⊂ R is a nilpotent ideal. 
By transitivity and an easy induction, it suffices to assume I^2 =0. 
Next we apply a standard trick to reduce to the case that R→ R/I is split:
choose a simplicial resolution P_∙→ R/I by polynomial ℤ-algebras (possibly in infinitely many variables), and let Q_∙ be
the fiber product along R→ R/I. Then each Q_j→ P_j has kernel I and
admits a section, since P_j is a polynomial algebra. Taking the geometric
realization using <Ref>(2), we thus reduce to proving
(<ref>) for each pair (Q_j=P_j⊕ I,I).
But we can write P_j as a filtered colimit of polynomial ℤ-algebras
on finitely many variables and I as a filtered colimit of finitely generated
modules. 
Using <Ref>(2) again (which shows that F commutes with filtered
colimits), 
and <Ref>, we conclude (<ref>) for I ⊂ R
nilpotent. 


Second, we prove (<ref>) in case R is noetherian and  F-finite, and 
R is I-adically complete. 
In this case, 
<Ref> shows that 
F(R) ≃ F(R/I^n). 
We consider the tower (in n), 
T_n = fib( F(R/I^n) → F(R/I)); the fiber of each successive map
T_n → T_n-1 belongs to D^≤ i( 𝔽_p), and thus we get that
_n T_n = fib( F(R) → F(R/I)) ∈ D^≤
i(𝔽_p) as desired. 


Finally, suppose (R, I) is a general henselian pair; we prove (<ref>).
Since F commutes with filtered colimits, it suffices to assume that the
pair (R, I) is the
henselization of a finitely generated ℤ-algebra R_0 along an ideal
I_0 ⊂ R_0. 
By the previous paragraph, we have
fib( F( R̂_I) → F(R/I)) ∈ D^≤ i(𝔽_p), since
R̂_I is an F-finite, noetherian ring. 
Now R_0 is an excellent ring as a finitely generated ℤ-algebra; since R_0 → R is ind-étale, R is also excellent
<cit.>. 
It follows that R →R_I is geometrically regular <cit.>
and is therefore a filtered colimit of smooth maps 
by Néron–Popescu desingularisation <cit.> <cit.>); each of these maps necessarily admits a section. 
In particular, 
the map 
fib( F(R) → F(R/I)) →fib( F(R_I) → F(R/I))
is a filtered colimit of maps, each of which admits a section. 
Since we have just seen that the target of this map belongs to D^≤
i(𝔽_p), the source does 
as well. 


To complete the proof of <Ref> by treating the pairs in the statement of  <Ref>, we will exploit the presence of the grading induced by the variables.

 
In the remainder of this subsection we will systematically use graded objects
indexed over a commutative monoid M (which will be ℤ[1/p]_≥0 or
ℤ_≥0): an M-graded object in a (∞-)category
𝒞 is
by definition a functor R_⋆  M→𝒞. When 𝒞 is
symmetric monoidal, then so is the resulting category
Fun(M,𝒞) of M-graded objects, under the Day convolution product, and an M-graded ring is then a (𝔼_∞-, etc.) monoid object in M-graded objects. 

The underlying object R of a graded object R_⋆ is by definition
⊕_m∈ MR_m, when it exists. When all direct sums exist, the functor
R_⋆↦ R is conservative (and faithful when 𝒞 is
an ordinary category).
We will be particularly interested in p-complete graded rings, by which
we mean a graded ring R_⋆ in the category of p-complete abelian groups;
the underlying object R is then the p-completed direct sum
⊕_m∈ MR_m, which is itself a p-complete ring. We
sometimes abusively identify ⊕_m∈ MR_m with R_⋆
itself; this is a mild abuse of notation given that the functor
R_⋆↦⊕_m∈ MR_m is conservative and faithful.

An M-graded commutative ring R_⋆ may of course be viewed  as an M-graded
𝔼_∞-ring in spectra, i.e., an 𝔼_∞-monoid in the
    symmetric monoidal stable ∞-category Fun(M,). By
    <Ref>, we may then
form the S^1-equivariant object (R_⋆)∈Fun(M,)^B S^1 and
the associated homotopy fixed points ^-(R_⋆), homotopy orbits
(R_⋆)_hS^1, and Tate construction (R_⋆), all of which are
M-graded spectra. Note that the underlying spectrum of (R_⋆)
is the  of
the underlying ring spectrum of R_⋆ 
because the underlying spectrum functor preserves tensor products and colimits;
this is not true for ^-, because the underlying spectrum functor need
not preserve limits (e.g., S^1-homotopy fixed points). 


[<cit.> for graded rings] 
Assume that the monoid M is uniquely p-divisible, such as ℤ[1/p]_≥0. Then the main constructions and results of <cit.> extend to M-graded rings. 


We will say that a p-complete M-graded ring R_⋆ is quasisyntomic (resp. quasiregular semiperfectoid) if the underlying ring R=⊕_m∈ MR_m is quasisyntomic (resp. quasiregular semiperfectoid). One has a natural graded analog of the quasisyntomic site, and similarly quasiregular semiperfectoids form a basis (for example, by extracting p-power roots of homogeneous elements as in <cit.>; this is why p-divisibility of M is required); one obtains an analog of unfolding in
this context. 

For such R_⋆, we have seen in <Ref> that we have
natural M-graded spectra,
(R_⋆; ℤ_p),  ^-(R_⋆;
ℤ_p), (R_⋆;
ℤ_p)_hS^1, and (R_⋆;
ℤ_p). Moreover, the latter are naturally filtered objects in M-graded p-complete spectra, by carrying over the construction of the motivic filtration of <cit.> to the graded context. 
It follows that we get M-graded p-complete objects
_R_⋆{i}, 𝒩^≥ i_R_⋆{i}, etc. 
In general, 
the underlying (p-complete) objects of  _R_⋆{i}, 𝒩^≥ i_R_⋆{i} 
do not agree with those of _R{i}, 𝒩^≥ i_R{i}, because the underlying object of ^-(R_⋆;
ℤ_p) is not ^-(R; ℤ_p). 
However, 
for each j ≥ i, the underlying p-complete object of 
𝒩^≥ i_R_⋆{i} / 𝒩^≥ j_R_⋆{i} is 𝒩^≥ i_R{i} / 𝒩^≥ j_R{i}. 
This follows because the underlying object of (R_⋆; ℤ_p)
is (R; ℤ_p) and the forgetful functor from M-graded spectra to
spectra commutes with finite homotopy limits. 

Throughout <cit.>, a basic tool is the cotangent complex and its 
wedge powers; here we implicitly use that if R_⋆ is a p-complete M-graded ring,
then we have natural p-complete graded objects ⋀^i
L_R_⋆/ℤ_p (defined in the usual manner as a left derived
functor of differential forms). The Hochschild–Kostant–Rosenberg theorem
remains valid in the graded context, and from there the results of
section <ref> carry over to this context as well. 
 

We now turn to the interaction of the Frobenius with the grading, in the case which interests us. 
 

Let R_⋆ be a quasisyntomic ℤ[1/p]_≥ 0-
(resp. ℤ_≥ 0-) graded ring (with underlying quasisyntomic ring R). 
Then we claim that


  * _R{i}/ 𝒩^≥ n_R{i} naturally
upgrades to have the structure of a ℤ[1/p]_≥ 0- (resp. ℤ_≥ 0-) graded object in the p-complete derived ∞-category D(ℤ_p)

  * the 
Frobenius map modulo p, as in (<ref>)

    φ_i𝒩^≥ i_R{i}/
    𝒩^i+r_R{i}⊗_ℤ^L𝔽_p →_R{i}/𝒩^≥ i+r_R{i}⊗_ℤ^L𝔽_p
 
 multiplies degrees by p. 



In the case in which R_⋆ is ℤ[1/p]_≥0-graded, part (1) is covered by the general <Ref>: arguing locally on the graded version of the quasisyntomic site, we require a natural graded version of <Ref> for any graded quasiregular semiperfectoid ring. But this follows from  of a graded ring being a graded spectrum with S^1-action, as explained in <Ref>.
 
When R is actually ℤ_≥ 0-graded, then we claim that the same is
true of _R{i}/ 𝒩^≥ n_R{i}. By
dévissage, it suffices to prove this for each 𝒩^t _R, which
in turn follows from the filtrations of <Ref> and
<Ref>. Here we use that the p-completed cotangent complex
L_R/ℤ_p and its wedge powers are naturally ℤ_≥ 0-graded, and the filtrations of the aforementioned corollaries respect
these gradings. 

For part (2) we apply a left Kan extension argument to assume that R is p-torsion-free, and then argue locally as above to reduce to the case that R is a p-torsion-free, graded, quasiregular semiperfectoid. Then both sides of ϕ_i are discrete, and so the problem reduces to verifying the property that it multiplies degrees by p. 
But this follows from the treatment of graded cyclotomic spectra
in <Ref>. 
 

 
Let R=A ⊕ N and I=N be a henselian pair of the form of
<Ref>. We view R as being ℤ_≥0-graded, with A
in degree 0 and N in degree one. We must prove that fib(𝔽_p(i)(R)→𝔽_p(i)(A))∈ D^≤ i(𝔽_p). Choose r≫0 so that expression (<ref>) is valid.

The degree 0 part of ⋀^i L_R/ℤ_p identifies with ⋀^i L_A/ℤ_p, so by dévissage using <Ref> and <Ref> we see that the same is true of the ℤ_≥ 0-graded object _R{i}/𝒩^≥ i+r_R{i}. Namely, its degree 0 part is _A{i}/𝒩^≥ i+r_A{i}. The same is true modulo p, and so fib(𝔽_p(i)(R)→𝔽_p(i)(A)) identifies with the fiber of 
    - ϕ_i (𝒩^≥ i_R{i}/
    𝒩^≥ i+r_R {i})_>0⊗_ℤ^L
    𝔽_p → 
    ( _R{i}/
    𝒩^≥ i+r_R {i})_>0⊗_ℤ^L
    𝔽_p,

where the subscript >0 denotes the ℤ_>0-subobject of a ℤ_≥0-graded object.

To complete the proof we must verify the conditions of <Ref> below. Firstly, the Frobenius multiplies degrees by p by <Ref>(2). Next, the fiber of can is (_R{i}/
𝒩^≥ i_R {i})_>0[-1], which lies in D^≤ i(ℤ_p) by <Ref>. Finally, to verify condition (2), observe that each cohomology group of ⋀^i
L_R/ℤ_p is a ℤ_≥0-graded, finitely generated R-module, so necessarily zero except in finitely many degrees of the grading. The same then holds for _R{i}/𝒩^≥ i+r_R{i} by another dévissage through <Ref> and <Ref>.


 

Let M, N be ℤ_>0-graded objects of D(
𝔽_p), and i≥0.
Let can M → N be a map of graded objects and let
φ M → N be a map which multiplies degrees by p. 
Suppose that

    
  (1)  the fiber of can belongs to D(𝔽_p)^≤ i,

  (2) 
for any fixed n, the cohomologies H^n(M), H^n(N) vanish except in finitely many degrees of the grading.

Then fib(  - ϕ M → N) belongs to
    D(𝔽_p)^≤ i.
 
 
By (2), we can replace the direct sums ⊕ M_i and ⊕ N_i with the
    corresponding infinite
products. Therefore, the result follows from <Ref>. 
 


Finally, we use the rigidity theorem to give a description of the top cohomology
of the 𝔽_p(i). 
For B an 𝔽_p-algebra, let Ω^i_B be the (underived) module of i-forms on B (relative to
ℤ_p, or 𝔽_p), and let 
    C^-1Ω^i_B→Ω^i_B/d Ω^i-1_B
 be the inverse Cartier operator.

 

Let R be a p-complete ring. 
Then there is a natural isomorphism
 
    H^i+1( 𝔽_p(i)(R)) ≃coker(1 -C^-1Ω^i_R/p→Ω^i_R/p/dΩ^i-1_R/p).

In particular, if R is w-strictly local, then H^i+1( 𝔽_p(i)(R)) =
H^i+1( ℤ_p(i)(R)) =0. 
 
 
Without loss of generality, we can assume that R is an 𝔽_p-algebra 
via <Ref>. 
We now use a standard argument (cf., e.g., <cit.>) to reduce to the case where R is ind-smooth
over 𝔽_p. 
Choose a polynomial 𝔽_p-algebra P surjecting onto R and henselize
P along the kernel of P ↠ R, forming a P-algebra P'
augmented over R; another use of <Ref> 
gives H^i+1(𝔽_p(i)(R)) = H^i+1( 𝔽_p(i)(P')). Moreover, 
<cit.> gives that 
the right-hand-side of 
<Ref> is also invariant under passage from P' to R. 
Replacing R by P', 
we thus reduce the case that R is ind-smooth over 𝔽_p. But then

    𝔽_p(i)(R) ≃fib( Ω^i_R Ω^i_R/d Ω^i-1_R ) [-i].
 
This follows because of the expression of 
𝔽_p(i) as the shifted étale cohomology of 
logarithmic forms
Ω^i_log
(cf. <cit.> and reduction modulo p; we also review this in the
next section) and the short exact
sequence of étale sheaves (cf. <cit.> and <cit.>)

    0 →Ω^i_log→Ω^i Ω^i/d Ω^i-1→ 0 .

Expression (<ref>) implies the claim. 
Note that this short exact sequence (in the étale topology) implies also that
for any ind-smooth 𝔽_p-algebra which has no non-split étale covers
(e.g., R could be w-strictly local), H^i+1( 𝔽_p(i)(R)) = 0. 
 


<Ref>,  <Ref>, and <Ref> imply Theorem G from
the introduction. 



§ THE COMPARISON WITH SYNTOMIC COHOMOLOGY









In this section, we show that the ℤ_p(i) for i ≤ p-2 and the
ℚ_p(i) for all i can be described purely in terms of derived de
Rham (instead of prismatic) cohomology, using a form of syntomic cohomology
<cit.>. 
The strategy is to use the description of the ℤ_p(i) in equal
characteristic p from <cit.> together with the Beilinson fiber square
to relate the ℤ_p(i) in mixed and equal characteristic. 
In particular, we prove Theorem [thm:f]F.




 §.§ Syntomic cohomology


To begin with, we define another form of syntomic cohomology 
via the quasisyntomic site, by descent from
quasiregular semiperfectoids. 

 
For a map of  rings A → R, we let L Ω_R/A∈ D(A) denote the
p-adic derived de Rham cohomology of R relative to A, see
<cit.>. 
By definition, when R is a finitely generated polynomial A-algebra, L
Ω_R/A is given by the p-completed relative de Rham complex
Ω_R/A^∙, and in general L Ω_R/A is defined via
p-complete left Kan extension. 
One can show <cit.> that L Ω_R/A more generally  agrees with
the p-completed (underived) relative de Rham complex when R is smooth
over A. When A = ℤ, we omit A from the notation. 

The p-adic derived de Rham cohomology L Ω_R/A is equipped with the derived Hodge filtration
L Ω_R/A^≥⋆, obtained by left Kan extending the
naive filtration in the polynomial (or more generally smooth) case. 

 

[Derived de Rham cohomology and divided powers] 

We recall the following basic calculation: for the map ℤ[x] →ℤ, we have that 
L Ω_ℤ/ℤ[x]≃Γ(x) is the
p-complete divided power algebra on the class x, and the derived Hodge filtration
is the divided power filtration. 
This is a special case of <cit.>. 
See also <cit.> for an account. 
 

 
For an 𝔽_p-algebra S, we let LWΩ_S ∈ D(
ℤ_p) denote the p-adic derived de Rham–Witt cohomology or
derived crystalline cohomology of S (defined via p-complete left Kan extension from
finitely generated polynomial 𝔽_p-algebras) <cit.>; for ind-smooth 𝔽_p-algebras S this agrees with Illusie's usual  WΩ_S. 
It comes equipped with the derived Nygaard filtration 𝒩^≥⋆LWΩ_S obtained via left Kan extension of the usual Nygaard
filtration in the finitely generated polynomial case (see <cit.> for an account);
we write LWΩ_S for the completion of LWΩ_S with respect to the Nygaard
filtration.
We have by <cit.> an identification of the associated graded terms 
of the Nygaard filtration 
𝒩^≥ iLWΩ_S/𝒩^≥ i+ 1LW Ω_S  ≃
L (τ^≤ iΩ_S/𝔽_p) with the derived functor of S
↦τ^≤ iΩ_S/𝔽_p. 
The Frobenius φ S → S induces an endomorphism of LW Ω_S; on
𝒩^≥ i L W Ω_S it becomes divisible by p^i, and indeed we
have divided Frobenius maps 
    φ_i: 𝒩^≥ i L W Ω_S → L W
    Ω_S.

Finally, using the de Rham-to-crystalline comparison, we find that if R is a
p-torsion free ring, then there is a natural equivalence

    L Ω_R ≃ L W Ω_R/p;
 
in particular, L Ω_R naturally carries a Frobenius operator φ. 
 


 

The functors S ↦ L W Ω_S and S↦LW Ω_S are  
sheaves on qSyn_𝔽_p. 
For LW Ω_S, it suffices to work modulo p, and then use the conjugate
filtration on derived de Rham cohomology <cit.> and the flat descent
for the wedge powers of the cotangent complex <cit.>. For the
Nygaard-completed  
LW Ω_S, this follows since the associated graded terms have
this property, by a similar argument. 
Similarly, the filtration pieces S↦𝒩^≥ i L W Ω_S
    and S↦𝒩^≥ iLW Ω_S are sheaves on
qSyn_𝔽_p. 
 




[Derived de Rham–Witt cohomology of quasiregular semiperfect
rings, cf. <cit.>] 

Let S ∈_𝔽_p be a quasiregular semiperfect
𝔽_p-algebra. In this case, one forms the ring (S) (defined by Fontaine <cit.>),
which is the universal p-adically complete divided power thickening of S,
with divided powers compatible with those on (p) ⊂ℤ_p;
quasiregularity ensures that it is p-torsion free. 
Then, one has a natural identification 
    LW Ω_S = (S),
 and the
Nygaard filtration becomes the filtration 

    𝒩^≥ i(S) = {x ∈(S)φ(x)∈ p^i(S)
    }.

Here φ(S) →(S) denotes the endomorphism induced by the
Frobenius on S; it has the further property that φ(x) ≡ x^p p for x ∈(S), i.e., φ defines the structure of a
δ-ring on the p-torsion free ring (S). 
 



[Derived de Rham and de Rham–Witt cohomology of qrsp rings] 

















Let R ∈_ℤ_p. 
We consider the following rings.


  * The derived de Rham–Witt cohomology LW Ω_R/p of R/p. 
Since the ring R/p is a quasiregular semiperfect 𝔽_p-algebra, 
it follows from Construction <ref> that 
there is an isomorphism

    LW Ω_R/p≃(R/p).


  * The (p-adic) derived de Rham cohomology L Ω_R. 
Here L Ω_R is a discrete, p-complete and p-torsion free ring, as follows from the previous point and (<ref>). The ring L Ω_R is also equipped with the multiplicative, descending
Hodge filtration L Ω_R^≥⋆. 

Via the de Rham-to-crystalline comparison, we have equivalences
L Ω_R ≃ LWΩ_R/p≃(R/p). 
In particular, we find via (1) and (2) above that the p-complete,
p-torsion free ring L Ω_R is equipped with both a Frobenius
operator and a Hodge filtration. 

 


 

Let r ≥ 0 be an integer. 


  (1) The p-adic valuation of (pr)!/r! is equal to r.  

  (2) The p-adic valuation of p^r/r! is at least min(r, p-1). 

 
 
Both assertions follow from Legendre's formula, v_p(n!) = ∑_j > 0⌊ n/p^j
⌋ for v_p the p-adic valuation. 
 
 
Let R ∈_ℤ_p. 
Then for i ≤ p-1, the Frobenius φ L Ω_R → L Ω_R carries L
Ω_R^≥ i into p^i L Ω_R, or in other words the de
Rham-to-crystalline comparison carries LΩ_R^≥ i into 𝒩^≥
i(R/p).
 
 
For any R ∈_ℤ_p, we can write R = W(A)/I, where A is a perfect
𝔽_p-algebra
and I ⊂ W(A) is an ideal. 
We have an identification of the p-complete cotangent complex,
L_R/ℤ_p≃I/I^2[1]. 


We first verify the 
assertion when the ideal I as above can be written as I = (f), for f a
nonzerodivisor, so R = W(A)/(f). 
In this case, 
in view of 
<Ref> and base change, 
we find that 
LΩ_R = L Ω_R/W(A) is the p-completion of 
the divided power envelope of the regular ideal (f), i.e., the ring
W(A)[ f^n/n!]_n ≥ 1; furthermore, for each i, the Hodge filtered piece L
Ω_R^≥ i identifies with the corresponding divided power filtration,
i.e., the ideal ( f^j/j!)_j
≥ i. 
Now the Frobenius φ on LΩ_R ≃ LW Ω_R/p≃(R/p) is a Frobenius lift coming from a δ-structure, so 

    φ( f^j/j!) =
    (f^p + p δ(f))^j/j! = ∑_0 ≤ l ≤ jjl f^pl p^j-lδ(f)^j-l/j!.

The lth term in the sum above is divisible (in the ring L Ω_R) by 
f^pl
p^j-l/l! (j-l)! =  f^pl/(pl)!(pl)!
p^j-l/l! (j-l)!, where we use the divided powers on (f) to see
f^pl/(pl)!∈ L Ω_R.
Now  the p-adic valuation of 
(pl)!
p^j-l/l! (j-l)!
is at least l + min(j-l, p-1) thanks to <Ref>. 
So if i ≤ p-1, 
then it follows that φ carries L Ω^≥ i_R into 
p^i L Ω_R. 

Now suppose R is a p-complete tensor product over α∈𝒜 of 
rings of the form W(A_α)/(f_α), for A_α perfect
𝔽_p-algebras and f_α∈ W(A_α) regular elements. 
In this case, we have an isomorphism (after p-completion) of filtered rings
L Ω_R^≥⋆≃⊗_α∈𝒜L
Ω_W(A_α)/f_α^≥⋆ by the Künneth formula, which is compatible with the
Frobenius operators. The assertion 
φ( L Ω_R^≥ i) ⊂ p^i L Ω_R for i ≤ p-1
for such R thus follows by taking tensor products. 

Finally, let R ∈_ℤ_p be arbitrary and write R =
W(A)/I for A a perfect 𝔽_p-algebra. 
To prove the claim φ(LΩ_R^≥ i) ⊂ p^i L Ω_R for i
≤ p-1, we will reduce to the previous cases, following the strategy of
<cit.>. 
Let {x_t}_t ∈ T be a system of generators for the ideal I
and for each t, we write
x_t = ∑_i ≥ 0 p^i [y_t, i] for some y_t, i∈ A. 
For each t ∈ T, we have a map 

    W(𝔽_p[u_1, u_2, …, ]_perf)/( [u_1] + p[u_2] +
    … ) → W(A)/I =  R

sending [u_i] ↦ [y_t, i]; note that the source belongs to
_ℤ_p, and its cotangent complex is the shift of a free of
rank 1 module. 
The map (<ref>) has image
on p-completed cotangent complexes given by the class of x_t. 

We consider the p-completed tensor product 
    R' def=W(A) ⊗̂⊗_t
    ∈ T ( W(𝔽_p[u_1, u_2, …, ]_perf)/ 
    [u_1] + p [u_2] + … 
    ),
 which maps surjectively to R (via the above maps) and induces a
surjection on H^0(L_-/ℤ_p[-1]).  
Comparing with the reduction mod p and using the Hodge and conjugate
filtrations on
derived de Rham cohomology, we find that 
LΩ_R'^≥ i→ L Ω_R^≥ i is a surjection for each
i. 
Since the previous discussion shows that ϕ( L Ω_R'^≥ i) ⊂
p^i L Ω_R' for i ≤ p-1, we can now conclude the claim for R by
naturality, as desired. 
 

Using the divisibility property of Frobenius, 
we can define, for R ∈_ℤ_p and i ≤ p-1, 
a divided Frobenius φ/p^i  L Ω_R^≥ i→ L Ω_R
(of discrete, p-torsion free abelian groups). 
Using the divided Frobenius, we now define syntomic cohomology; this definition 
is based on the ideas of <cit.> (and can be compared with it using
the comparison between derived de Rham and crystalline cohomology in the lci
case, cf. <cit.>). 
 
We define sheaves 
i for 0 ≤ i ≤ p-2, and i for i≥0, 
on _ℤ_p via 

    i(R) = fib( ϕ/p^i - 1 L Ω_R^≥ i→
        L Ω_R),
    i(R) = fib( ϕ - p^i  L Ω_R^≥ i→ L Ω_R)__p.

These are sheaves on 
_ℤ_p because R ↦ L Ω_R^≥ i
is a sheaf. 
Unfolding, we obtain sheaves i for 0 ≤ i ≤ p-2 and i for all
i≥0 on qSyn_ℤ_p. 
 


    While one could define _p(p-1)^FM(R) via the same formula, this does not
    give the correct integral theory in weight (p-1).
	 





 §.§ The ℤ_p(i) in  equal characteristic p


In equal characteristic p, the ℤ_p(i) can be determined via
the theory of the de Rham–Witt complex and its derived versions, cf.
<cit.>, <cit.>, and in particular
<cit.>.[See also <cit.> for the identification with  
with p-adic étale motivic cohomology.] 
We next review this. 

 

Suppose S is a quasisyntomic 𝔽_p-algebra. 
Then, for each i,


  (1) 
_S{i} is the Nygaard-completed derived de Rham–Witt cohomology
LWΩ_S
of S and

  (2)
the Nygaard filtration 𝒩^≥ i_S{i}
identifies with the de Rham–Witt Nygaard filtration 
𝒩^≥ iLWΩ_S, and the prismatic Frobenius
φ_i identifies with the divided Frobenius (<ref>). 

Consequently,
 
    ℤ_p(i)(S) = fib( φ_i - can𝒩^≥ iLWΩ_S →LWΩ_S),

where φ_i: 𝒩^≥ iLWΩ_S →LWΩ_S is the divided Frobenius operator, so that
p^iφ_i is the Frobenius. 


 

The Nygaard completion is 
redundant in the formula (<ref>) for  ℤ_p(i)(S). 
This follows easily from the fact that 
φ_i acts by zero  on 𝒩^≥ i+1LWΩ_S/p. 
In particular, we can write

    𝔽_p(i)(S) = fib( φ_i - can  
    (𝒩^≥ iLWΩ_S / 
    𝒩^≥ i+1LWΩ_S) ⊗_ℤ_p^L 𝔽_p  → (LW Ω_S/𝒩^≥
    i+1LWΩ_S) ⊗_ℤ_p^L 𝔽_p).


 
If S is
a quasiregular semiperfect 𝔽_p-algebra, then  for i  > 0,
ℤ_p(i)(S) is discrete, p-torsion free, and 
there is a natural  isomorphism of abelian groups

    ℤ_p(i)(S) ≃ker( ϕ  - p^i(S) →(S)),
 
where φ is induced by the Frobenius. For i = 0, we should instead take the homotopy fiber of φ - 1 on
(S), so it may have terms in cohomological degree 1. 
 

In the ind-smooth case, one has an 
identification with logarithmic de Rham–Witt forms. 

 
For S an ind-smooth 𝔽_p-algebra, we  let W Ω_S, log^∙  denote the graded subring 
of the de Rham-Witt complex W Ω_S^∙ consisting of fixed points for F. When S is local, one knows that W
Ω_S, log^∙ 
is generated, modulo any power of p, in degree 1 by elements of the form d[x]/[x], for x ∈
S^× and [x] ∈ W(S) the Teichmüller representative, cf. <cit.> which proves this étale locally and <cit.> for a very general Zariski local result. 
Note that for each i, W Ω_-, log^i defines a pro-étale
sheaf on (S).
 


 

Let S be an ind-smooth 𝔽_p-algebra. 
Then there are natural identifications

    ℤ_p(i)(S) ≃ R Γ_proet ( (S), W Ω_-,
    log^i)[-i].

 





 §.§ The Beilinson fiber square on graded pieces


Our goal is to relate the ℤ_p(i) in mixed and in equal
characteristic. 
We use the Beilinson fiber sequence to prove a basic fiber square which gives a version of Theorem [thm:a]A on associated
graded terms for the motivic filtrations.
[The trace on graded pieces] 
Let R be any commutative ring. Then we have the trace maps 

    (R ; ℤ_p) →TC(R; ℤ_p) →^-(R;
    ℤ_p).

 When R ∈_ℤ_p, 
then ^-(R; ℤ_p) is concentrated in even degrees and 
π_2i is given by L Ω_R^≥ i, cf. <cit.>
and <cit.>. 
Unfolding we conclude that, on graded pieces, we obtain a natural map
ℤ_p(i)(R) →L Ω_R^≥ i
for R∈qSyn_ℤ_p. This naturally factors through L Ω_R^≥ i since R ↦ℤ_p(i)(R) is left Kan extended from p-complete polynomial algebras
(<Ref>). 
 


 

Let R ∈qSyn_ℤ_p.  Then, for each i ≥ 0, there exists a natural 
map χ_iℚ_p(i)(R/p) → (LΩ_R)_ℚ_p
and a functorial pullback square

    ℚ_p(i)(R)  [d]  [r]    ℚ_p(i)(R/p) [d]^χ_i
    
    (LΩ_R^≥ i)_ℚ_p[r]     (L
    Ω_R)_ℚ_p

in the  derived ∞-category D(ℚ_p).
The map χ_i arises from a natural map ℤ_p(i)(R/p) → p^-N L
Ω_R for some N ≫ 0 (depending only on i), fitting into an analogous commutative diagram.

Furthermore, the associated fiber sequence holds up to isogeny: 
cofib(ℤ_p(i)(R) →ℤ_p(i)(R/p)) 
and L Ω_R/L Ω_R^≥ i are naturally isogenous to each other. 
Finally, 
for i ≤  p-2, we have natural equivalences for
qSyn_ℤ_p,

    fib(ℤ_p(i)(R) →ℤ_p(i)(R/p)) 
    ≃fib(
    L Ω_R/L Ω_R^≥ i→ 
    L Ω_R/p/L Ω_R/p^≥ i)[-1]
    .
 

 
Note that the hypothesis that R ∈qSyn_ℤ_p ensures that R/p ∈. 
The square (<ref>) will be constructed in 
the ∞-category of D(ℚ_p)^≥ 0-valued sheaves on
qSyn_ℤ_p. 
It suffices to construct the above pullback square for  R ∈_ℤ_p, by
unfolding. 
For R ∈_ℤ_p, we have a pullback square by
<Ref>,

    (R; ℚ_p) [d]  [r]    (R/p; ℚ_p) [d]  
    ^-(R; ℚ_p) [r]    (R; ℚ_p).

Note  since R
∈_ℤ_p, 
the terms in the bottom row of the  above fiber square  are 
 concentrated in even degrees
<cit.>. 
Consequently, for each i, we can apply 
τ_[2i-1, 2i] and still obtain a fiber square. 
By definition of the ℚ_p(i) and by 
the corresponding description  of derived de Rham cohomology, as in <cit.> (or using the filtration of <cit.>), we obtain
(<ref>) for R (after a shift), albeit with a Hodge completion. 
In particular, instead of χ_i, we obtain a completed version

    χ̂_̂îℚ_p(i)(R/p) → (L
    Ω_R)_ℚ_p,

as well as a Hodge-completed version of the fiber square (<ref>). 

We can refine χ̂_̂î to χ_i (and obtain (<ref>)) as follows.  
First, by construction of these maps via the Beilinson fiber square, a multiple of χ̂_̂î lifts to a map 
ℤ_p(i)(R/p) →L Ω_R. Since the source is left Kan
extended (as a functor to the p-complete derived ∞-category) from
finitely generated p-complete polynomial algebras, we can restrict and left Kan extend to obtain that a
multiple of χ̂_̂î lifts to ℤ_p(i)(R/p) → L Ω_R. 
Inverting p, we obtain that χ̂_̂î factors through a map 
χ_iℚ_p(i)(R/p) → (L Ω_R)_ℚ_p. 

From the quasi-isogeny between (R, (p); ℤ_p) and 
Σ(R, (p); ℤ_p) as in <Ref>, 
 we
obtain the isogeny claim.


Finally, we verify (<ref>); again, we can assume that R ∈_ℤ_p by unfolding. 
Since everything is a pro-étale sheaf, we can even assume that R is
w-strictly local in the sense of <cit.>, so that π_-1(R;
ℤ_p) = coker(F - 1 W(R) → W(R)) 
(by <cit.>)
vanishes. 
Recall we have an equivalence
τ_≤ 2p-4TC(R, (p); ℤ_p) 
≃τ_≤ 2p-4ΣHC(R, (p); ℤ_p) by <Ref>. 
It follows that for i ≤ p-2, we have an equivalence

    τ_[2i-1, 2i]TC(R, (p); ℤ_p) ≃τ_[2i-1, 2i]ΣHC(R, (p); ℤ_p).

Now TC(R/p; ℤ_p), HC(R; ℤ_p), and
HC(R/p; ℤ_p)
are concentrated in even degrees since R ∈_ℤ_p. For the first claim see
<cit.>. 
The second and third follow from the filtrations
constructed in <cit.> and <cit.>. 

It follows from the above definitions that 

    τ_[2i-1, 2i]TC(R, (p); ℤ_p) ≃fib( ℤ_p(i)(R) →ℤ_p(i)(R/p))[2i],

 and from <cit.> 
and <cit.> 
 that 

    τ_[2i-1, 2i]ΣHC(R, (p); ℤ_p) 
    ≃fib(L Ω_R/L Ω_R^≥ i→ L
        Ω_R/p/L Ω_R/p^≥ i)[2i-1].

Using these identifications, 
we deduce (<ref>). 
 





We next identify the p-adic Chern character 
χ_iℚ_p(i)(R/p) → (L Ω_R)_ℚ_p on graded pieces
(from <Ref>)
more
explicitly. 
To this end, we prove the following basic result: 

 

Let R ∈_ℤ_p. 
Then for each i > 0, the map (of ℚ_p-vector
spaces) χ_iℚ_p(i)(R/p) → (L
Ω_R)_ℚ_p = (R/p)_ℚ_p is injective, and has
image given by the ϕ = p^i eigenspace. 
 
The main issue is the following: both the source and target of
χ_i are functors of R/p, thanks to de Rham–Witt theory. We have seen that the p-adic Chern character χ_i induces a natural map 
ℤ_p(i)(R/p) → p^-N L Ω_R for R ∈qSyn_ℤ_p
for some N. 
However, it is not a priori obvious that the map χ_i arises from a natural
transformation of functors on 𝔽_p-algebras (which would force it to
commute with Frobenius operators, for example). Our first goal is to verify
this. 





For any R ∈qSyn_𝔽_p, the Frobenius on R acts as multiplication by p^i on
ℤ_p(i)(R). 
 
 
This reduces to the case of a  quasiregular semiperfect
𝔽_p-algebra by descent. But in this case, the identification of <Ref> clearly proves
the claim.
 

 

The natural map χ_iℤ_p(i)(R/p) → p^-N(L Ω_R) → (L
Ω_R)_ℚ_p, for R
∈qSyn_ℤ_p  arises (by precomposition with reduction mod p)
from a unique natural transformation
χ_iℤ_p(i)→ p^-N' L W Ω_(-) on
qSyn_𝔽_p for some N' ≥ N. 
 
 
This follows from 
<Ref> and <Ref> (the latter shows that the
hypotheses of the former are satisfied), and then left Kan extension from
finitely generated p-complete polynomial rings. 
Uniqueness follows since these sheaves are torsion free. 
 


Next, we consider the sheaf of graded 𝔼_∞-rings
⊕_i = 0^∞ℤ_p(i) on
qSyn_𝔽_p. 
For each N ≥ 0, we can also truncate to 
obtain a sheaf of graded 𝔼_∞-rings ⊕_i = 0^N
ℤ_p(i). 


Let f⊕_i = 0^N ℤ_p(i) →⊕_i = 0^N
ℤ_p(i) be a natural map of sheaves of graded 𝔼_∞ rings on
qSyn_𝔽_p. Then there exists λ∈ℤ_p such that
in degree i, f is given by multiplication by λ^i. 
 

 
We first observe that the only endomorphisms of ℤ_p(1) (as a
functor on qSyn_𝔽_p) are given by
scalars. 
It suffices to verify this on quasiregular semiperfect algebras, and there
ℤ_p(1) is corepresentable (cf. <cit.>) by  𝔽_p[x^1/p^∞]/(x-1),
on which ℤ_p(1)( 𝔽_p[x^1/p^∞]/(x-1)) ≃ℤ_p. 
So the endomorphism f is given by a scalar action at least on
ℤ_p(1). 

Note that all these functors are left Kan extended from smooth algebras (to
the p-complete category), so 
f is determined by the values on smooth 𝔽_p-algebras. 
Furthermore, 
the map f
is determined by its values modulo p^n for each n. 
However, classes in H^i( ℤ/p^n(i)) are étale locally written as
sums of products of classes in H^1( ℤ/p^n(1)) (thanks to
<Ref>), so the value of f
on ℤ_p(1) determines the value of f in general. The result now
follows because on smooth algebras, ℤ/p^n(i) is concentrated in
cohomological degree i étale locally. 
 

 
Recall that the map χ_i is actually a special case of a map 
ℤ_p(i)(R_0) → p^-N'(L W Ω_R_0)
defined on R_0 ∈qSyn_𝔽_p, by <Ref>. 
For R_0 quasiregular semiperfect, 
we know that the Frobenius acts as p^i on ℤ_p(i)(R_0), so 
we obtain a natural, multiplicative map 
ℚ_p(i)(R_0) → ((R_0)^ϕ = p^i)_ℚ_p. 
We wish to see that these maps are isomorphisms. 

Now we know independently that ℚ_p(i)(R_0) is identified (for i
> 0) with 
(R_0)^ϕ = p^i_ℚ_p via the theory of topological cyclic homology
(<Ref>, following <cit.>). 
Thus, we actually obtain natural, multiplicative (in i) maps 
ℚ_p(i)(R_0) →ℚ_p(i)(R_0)
for R_0 ∈qSyn_𝔽_p, and we wish to see that these are
isomorphisms. Up to rescaling by a power of p, furthermore, they carry
ℤ_p(i)(R_0) into ℤ_p(i)(R_0). 
As we saw in 
<Ref>, these maps are necessarily all given by scalar multiplication by
some λ^i in degree i, for some λ∈ℤ_p;
we know that λ≠ 0 (by comparing with i =1, say), so the
result now follows.  



 §.§ Comparison of the i and ℤ_p(i)

Our main result is the following comparison, which establishes Theorem [thm:f]F. 
 

For R∈qSyn_ℤ_p, there are natural, multiplicative identifications
i(R) ≃ℤ_p(i)(R) for i ≤ p-2 and
i(R) ≃ℚ_p(i)(R) for all i≥0.
 

By <cit.>, for i ≤ p-2 and for formally smooth schemes
over DVRs, syntomic cohomology in the above form (see also <cit.>) is p-adic étale motivic cohomology. 


Fix i≥0. It is enough to prove the equivalences for all R ∈_ℤ_p. Thanks to the odd vanishing conjecture proved in
<cit.>, we may
moreover assume that R ∈_ℤ_p is such that
ℤ_p(i)(R) is concentrated in degree zero. In the homotopy cartesian
    square of <Ref>, the terms ℚ_p(i)(R), (L Ω^≥
    i_R)_ℚ_p, and (L Ω_R)_ℚ_p are all
    concentrated in degree 0, whence the same is true of the remaining term
    ℚ_p(i)(R/p) (that is, φ - p^i (L
    Ω_R)_ℚ_p→ (L Ω_R)_ℚ_p is surjective)
    and the fiber square is simply a cartesian and cocartesian square of
    abelian groups

    ℚ_p(i)(R) [d]  [r]    ℚ_p(i)(R/p) [d]  
    
    (L Ω^≥ i_R)_ℚ_p[r]      (L Ω_R)_ℚ_p.

Note that all the arrows are injections: the bottom since it is the inclusion of the Hodge filtration, the right by <Ref>, and the others since the diagram is cartesian.


We claim that the map ϕ - p^i (L Ω_R^≥ i)_ℚ_p→ (L
Ω_R)_ℚ_p is surjective. 
Indeed, 
given x ∈ (L Ω_R)_ℚ_p, we can write x = (ϕ - p^i)(x')
for some x' ∈ (L Ω_R)_ℚ_p; as we noted above, ϕ - p^i (L Ω_R)_ℚ_p→ (L Ω_R)_ℚ_p is surjective. Using <Ref> to identify the image of the vertical map, the diagram being cocartesian means that
(L Ω_R)_ℚ_p^ϕ = p^i⊕ (L Ω_R^≥
i)_ℚ_p→ (L Ω_R)_ℚ_p = (R/p)_ℚ_p
is surjective. So we can write x' = y' + z' for y' ∈(
φ - p^i) and z' ∈ (L Ω_R)_ℚ_p^≥ i. 
Applying φ - p^i, we get that x = (φ - p^i)(z'), proving the
claim as desired. 

Combining these observations, we have established a natural  identification

    ℚ_p(i)(R) = (L Ω_R)_ℚ_p^ϕ = p^i∩ (L Ω_R^≥
    i)_ℚ_p≃fib( φ - p^i (L Ω_R^≥
    i)_ℚ_p→ (L Ω_R)_ℚ_p),

as desired.
 


 
Let R be any simplicial commutative ring. Then there is a natural equivalence

    (R; ℚ_p) ≃⊕_i ≥ 0fib( φ -
    p^i L Ω_R^≥ i→ L Ω_R )_ℚ_p.

 
 
The map from R to its derived p-adic completion induces an equivalence on all the terms appearing in the statement: for derived de Rham cohomology and its Hodge filtration this follows from base change, while it holds for (-;ℤ_p) (and hence (-;ℚ_p)) by <cit.>. We may therefore assume R is p-complete, at which point we know from <Ref> that (R;ℚ_p) admits a complete descending filtration with associated graded given by ℚ_p(i)(R)[2i], for i≥0. Using Adams operations on 
as in <cit.>, we can
split the filtration functorially. 
Combining with the rational part of <Ref> (or more precisely
its left Kan extension to p-complete simplicial commutative rings), the claim follows.
 




Next, we will prove the integral case of <Ref>. 
The main step is to show that the i(-) for i≤ p-2  are discrete, as sheaves on qSyn_ℤ_p;
this is the analog of <Ref>, i.e., of the odd vanishing conjecture. 
To see this, we will use the odd vanishing conjecture itself and some cases of
the results of Li–Liu <cit.>. 

 

As D(ℤ_p)^≥ 0-valued sheaves on qSyn_ℤ_p,

    
  (1) i(·) is discrete and torsion-free for 0≤
        i≤ p-2 and
    
  (2) i(·) is discrete. 

 
 
Item (2) has already been proved above (in light of the odd vanishing
conjecture <cit.>), so here we prove (1).  
It suffices to show that i(·) is discrete and torsion-free as a
D(ℤ_p)^≥ 0-valued sheaf on qSyn_𝒪_C, for
𝒪_C the ring of integers in C = ℂ_p. 
In particular, we will show that for any R ∈qSyn_𝒪_C, there exists a quasisyntomic cover R → R' such that 
R' is quasiregular semiperfectoid and such that 
ϕ/p^i - 1  L Ω_R'^≥ i→ L Ω_R' is surjective
for all i ≤ p-2. 

To this end, on 
_𝒪_C, we consider the (non-Nygaard-complete) sheaf
R ↦_R (cf. <cit.>) together with its Nygaard filtration 𝒩^≥⋆_R and divided Frobenius maps ϕ/ξ^i𝒩^≥ i_R →_R, for ξ
a generator of the ideal defining the prism structure
on _𝒪_C. 
For any R ∈_𝒪_C, we use <cit.> (applied to the perfect prism ((𝒪_C),
ξ))  
or the crystalline comparison 
<cit.>
to obtain a natural map 
of δ-rings

    _R →(R/p) ≃ L Ω_R.

This map of δ-rings 
carries ξ to an element of the form p ϕ(u), for u ∈ (L
Ω_R)^× and ϕ L Ω_R → L Ω_R the crystalline Frobenius. 
By <cit.>, 
the map 
(<ref>) canonically refines to a filtered map
𝒩^≥⋆_R→  L Ω_R^≥⋆. 
Moreover, the filtered map 
𝒩^≥⋆_R → L Ω_R^≥⋆ induces an
isomorphism on associated graded terms in degrees ≤ p-1 (loc. cit.). 

By the odd vanishing conjecture <cit.>, there exists a basis of
objects R ∈_𝒪_C for which the map
ϕ/ξ^i -1 𝒩^≥ i_R →_R is surjective. Choose any such R.  Consider now the evident commutative diagram

    𝒩^≥ i_R [d]  [r]^ϕ/ξ^i-1   _R [d]  
    
    L Ω_R^≥ i[r]^ϕ/ξ^i -1    L Ω_R
    .

For i ≤ p-2, it follows from 
the surjectivity of the top horizontal arrow in 
(<ref>) (and that ξ maps to p ϕ(u) in L
Ω_R, for u ∈ (L Ω_R)^×) 
that the composite 

    L Ω_R^≥ i  L Ω_R → L Ω_R/L
    Ω_R^≥ i+1≃_R/𝒩^≥ i+1_R

is surjective. 
Now the map 

    L Ω_R^≥ i + 1⊂ L Ω_R^≥ i
    L Ω_R → L Ω_R / p

is simply the inclusion composed with reduction mod p, since ϕ/p^i is
divisible by p on  
L Ω_R^≥ i + 1. 
This observation combined with the surjectivity of (<ref>) and
p-completeness now gives the surjectivity of L Ω_R^≥ i L Ω_R as desired. 



Fix i ≤ p-2 and R ∈_ℤ_p such that ℤ_p(i)(R) and i(R)
are concentrated
in degree zero for all i ≤ p-2; we can do this by the odd vanishing
conjecture and <Ref>. 
In this case, the map of discrete abelian groups ℤ_p(i) (R) →ℤ_p(i)(R/p)
is injective and has torsion free cokernel, thanks to the equivalence
(<ref>). So from the Beilinson fiber square on graded pieces (<Ref>) and the description of the image of χ_i (<Ref>), we find that ℤ_p(i)(R) ⊂ℤ_p(i)(R/p) = (L
Ω_R)^ϕ = p^i is
the submodule 
consisting of those elements such that the image in L Ω_R belongs to
L Ω_R^≥ i. 
In particular, it is precisely the kernel of φ/p^i -1  L Ω_R^≥
i→ L Ω_R. Since this map is surjective, we get the natural 
equivalence ℤ_p(i)(R) ≃ℤ_p(i)(R)^FM as
desired. 






§ EXAMPLES




 §.§ K-theory of p-adic fields


Let F be a complete discretely valued field of characteristic 0 with ring
of integers _F ⊂ F and 
perfect residue field k of characteristic p. In this subsection, we will use
the Beilinson fiber sequence to recover various calculations of the p-adic
-theory of F. All these results are previously known at least in the case of F local; see
<cit.> for a detailed survey.



The homotopy groups of (F;_p) are given (as ℚ_p-vector spaces) as follows: 


  * _2s(F;_p)=0 for s>0;

  * there is a natural isomorphism _2s-1(F;_p)≃ F for each s>1;

  * there is a natural short exact sequence 0→ F→_1(F;_p)→_p→ 0.
 


Since k is perfect, we have that _i(k; ℤ_p) = ℤ_p for
i = 0 and 0 otherwise, cf. <cit.> and <cit.>. 
Taking the dévissage cofiber sequence (k)→(_F)→(F) with _p-coefficients shows that _i(_F;_p)_i(F;_p) for i≠ 1 and that there is an exact sequence 
    0→_1(_F;_p)→_1(F;_p)→_p→
        0,

where the map _1(F;_p) F^×⊗__p→_p is induced by 
    the p-adic valuation.
    
Next, since (_F/p;_p)(k;_p)_p, the Beilinson fiber
square (Theorem [thm:a]A) for _F yields a fiber sequence

    Σ(_F;_p)→(_F;_p)→_p.
 But
note that the cyclic homology term may be equivalently written as Σ(F/F_0),
where F_0:=W(k)[1p]; indeed, the vanishing of the p-adic completion of
the cotangent complex L_W(k)/ implies that (_F;_p)≃(_F/_F_0;_p), but (_F/_F_0) is already
derived p-adically complete since its homology groups are all finitely
generated _F-modules. The Beilinson fiber sequence therefore implies
that 
    τ_≥2Σ(F/F_0)≃τ_≥2(F;_p)
 and

    _0(F/F_0)≃_1(_F;_p).
 The proof is completed by noting
that, since F is an étale F_0-algebra, its cyclic homology is given by
_i(F/F_0)=F for i≥0 even and is 0 otherwise.


[Local fields]
If F is a finite extension of _p, of degree d, the theorem shows that 

    __p_2s-1(F;_p)=d+1   if s=1,
    
    d   otherwise.
 This dimension calculation is a
classical result, arising from Wagoner's <cit.> calculation of the
ranks of the continuous -groups and Panin's <cit.> proof of an early
case of the p-adic continuity of -theory.

In addition, the dimension calculation (<ref>) is in accordance with the Beilinson–Lichtenbaum conjecture for
F. Recall that the Beilinson–Lichtenbaum conjecture, prior to its
general proof by Rost–Voevodsky for all fields, was proved by Hesselholt–Madsen
<cit.> in this case when p > 2 using TC-theoretic methods. 
Since F has cohomological dimension 2, 
the Beilinson–Lichtenbaum conjecture 
predicts
_2s-1(F; ℚ_p) ≃ H^1_(F, ℚ_p(s)) and
_2s-2(F; ℚ_p) ≃ H^2_(F, ℚ_p(s)) for s > 0. 
Then the dimensions in (<ref>) agree with the dimensions of the
ℚ_p-cohomology of F, as computed via Tate's local duality and Euler characteristic
formula 
(see <cit.> for an account). 


[Integral calculation, unramified case]
Assume in this example that p> 3 so that the results hold in a non-empty
range. We will show that, in the range 1≤ i≤ 2p-5, the p-adic -groups of W(k) are given by 

    _i(W(k);_p)≃W(k)    if i=2s-1,
    
     0    if i is even.
 
Note that for k finite, the calculation of the entire homotopy type of ( W(k); ℤ_p)
is carried out by Bökstedt–Madsen <cit.> at odd primes and Rognes
<cit.> at p= 2 (for k = 𝔽_2); again, see  
<cit.> for a survey for all of these results. 
	 
	 The integral form of the Beilinson fiber sequence (Corollary
    [quasiisogenythmB]B) takes the form of a natural fiber sequence 

    τ_≤ 2p-5Σ(W(k),(p);_p)→τ_≤ 2p-5(W(k);_p)→_p.
 As in the proof of Theorem <ref>, the cyclic homology term may be replaced by the cyclic homology  τ_≤ 2p-5Σ(W(k),(p)/W(k)) over W(k). 

A standard calculation of derived de Rham cohomology with divided powers (as in
<cit.>) gives L Ω_k/L Ω^≥ s_k
≃ W(k)/p^s for s≤ p-1; in general, L Ω_k/L Ω^≥ s_k is
discrete for all s. 
Using the filtrations of <cit.>, we conclude
for i≤ 2p-1,

    π_i(k/W(k))≃
            W(k)/p^s+1   if i=2s≥ 0,
    
        0   otherwise.

See also <cit.> for this calculation.
Also, _i(W(k); ℤ_p) _i(W(k)/W(k)) W(k) for i≥0 even and
is
zero otherwise. Thus, in the range 0≤ i≤ 2p-2, we deduce that
    
    π_i((W(k),(p)/W(k))≃p^s+1W(k)    if i=2s, 
     0    if i is odd
 This completes the proof.



    Assume now that F is ramified so that _F/p k[x]/(x^e), where e
    is the absolute ramification degree of F.
    Corollary [quasiisogenythmB]B implies,
    by rewriting the p-adic cyclic homology as non-completed cyclic
    homology with respect to W(k), that τ_≤
    2p-5Σ(_F,(p)/W(k))≃τ_≤
    2p-5(_F,(p);_p).

We now appeal to the fact that the algebraic -theory of truncated
polynomial rings over fields is
known <cit.>. The positive even
p-adic -groups of k[x]/(x^e) vanish so that we get



5-term exact sequences

    0→π_2s-1Σ(_F,(p)/W(k))→_2s-1(_F;_p)
    →_se(k)/V_e_s(k)→π_2s-2Σ(_F,(p)/W(k))→_2s-2(_F;_p)→ 0

for 2≤ s≤ p-2. In low degrees, this gives a computation of the integral
p-adic -groups of _F which is independent of
 <cit.>; on the other hand, using this calculation and
<cit.>, we can view the computation as giving information about the low-degree
étale cohomology of F.


We can also carry out previous types of calculations for the syntomic complexes
ℤ_p(i) rather than -theory. 


    Let _F be a complete discrete valuation ring of mixed characteristic
    (0,p) with perfect residue field k.

    
  (1) We have natural identifications

    _p(i)(_F)≃R
    Γ_proet( (k), _p )   if  i=0,
    
    F[-1]    if i>0.


  (2)
    In the unramified case _F = W(k),

    _p(i)(W(k))
    ≃ 
    RΓ_proet((k), _p)    if  i = 0,
     
    W(k)[-1]    if  0 < i ≤ p-2.
 

 
 
For (1), 
we have (where the first equivalence follows from <Ref> and taking
powers of Frobenius)

    _p(i)(_F/p) ≃ℚ_p(i)(k) ≃
           R Γ_proet( (k), ℚ_p) 
    		     if i=0,
    
            0   otherwise.

    Therefore, the fiber sequence from <Ref> yields equivalences
	 
    _p(i)(_F)≃
    	R Γ_proet((k), ℚ_p) 
    	    if  i=0,
    (L
    	 Ω__F/LΩ^≥ i__F)__p[-1]    if
    	 i>0.
 As in the proof of Theorem <ref>,
	 the latter truncated p-adic derived de Rham cohomologies may be computed
	 as the analogous un-completed derived de Rham cohomologies for F_0→ F;
     since L_F/F_0≃ 0, we conclude that _p(i)(_F)≃
     F[-1] for i>0.

The integral claim follows from 
<Ref>. 
Indeed, 
we find that ℤ_p(0)(W(k)) = fib( φ-1 W(k) → W(k))
≃ R Γ_proet( (k), ℤ_p). 
For i > 0, 
we get ℤ_p(i)(W(k)) = fib( φ/p^i - 1 0 → W(k)), so
the claim follows. 
 






 §.§ Perfectoid rings

In this section, we apply the Beilinson fiber square to a perfectoid ring. 
The main result (which was indicated to us by Scholze) is that 
it recovers the fundamental exact sequence in p-adic Hodge theory. 

Let R be a perfectoid ring. We review the period rings associated to R and their interpretation via
derived de Rham theory, cf. <cit.>. 


[Period rings] 

Let R be a perfectoid ring. 


  * As before, we have Fontaine's ring (R) equipped with the canonical map
θ(R) → R with kernel (ξ). Here (R) is also the prismatic cohomology
_R; the Nygaard filtration is the ξ-adic filtration. 

  * We have (R) = (R/p), the p-adic completion of the divided power envelope of (ξ) ⊂(R); we
have (R) ≃ L Ω_R is the derived de Rham cohomology of R. 
The Hodge filtration is given by the divided power filtration. 
We let (R) =  (R)[1/p] = (L Ω_R)_ℚ_p; the ring
(R) also inherits a Frobenius operator φ. 

  * We have ^+(R) =  ((R)/ξ^n [1/p]). 
The ring ^+(R) can also be obtained as the Hodge completion of (L
Ω_R)_ℚ_p.  
The Hodge filtration yields a filtration 
(the ξ-adic filtration) on ^+(R). 

 
Our goal is now to recover the following result in p-adic Hodge theory,
cf. <cit.> and <cit.>.


 

For any R ∈ and i>0, there is a natural pullback square
in D(ℚ_p),

    Γ_proet((R[1/p]), ℚ_p(i)) 
    [d]  [r]    (R )^φ = p^i[d]  
    Fil^≥ i^+(R) [r]    ^+(R) 
     
    .

 

 
    When R=_C, where C is a complete algebraically closed
    nonarchimedean field, the fundamental exact sequence is often written as the
    exact sequence

    0 →ℚ_p(i) →(𝒪_C)^φ = p^i→^+(𝒪_C)/Fil^≥ i^+(𝒪_C) → 0

of abelian groups.
 

 
We apply <Ref> to the perfectoid ring R and obtain a fiber
square

    ℚ_p(i)(R) [d] [r]    ℚ_p(i)(R/p) [d] 
    
    (L Ω_R^≥ i)_ℚ_p[r]     
    (L Ω_R)_ℚ_p.

By <cit.>, the first term is identified with
Γ_proet( (R[1/p]),
ℚ_p(i)). 
The ring R/p is quasiregular semiperfect, so we have
ℚ_p(i)(R/p) ≃(R)^φ = p^i
<cit.>. 

Note that we can replace the square (<ref>) by Hodge completing the
bottom row and
it will still remain cartesian, since the homotopy fibers do not change. 
This yields a new homotopy cartesian square
where one identifies the rings as in <Ref>, and then the result
follows. 
 
 



 
Unfortunately, in general we do not know a good way of identifying the map
(𝒪_C/p;_p) →HP(𝒪_C; ℚ_p) with the usual map in the fundamental
exact sequence. However, we can argue that it has to match with the
usual map, at least for C = ℂ_p, by appealing to some general results. 
For simplicity, in this example, we drop the argument of the perfectoid ring,
i.e., we write ^+ for ^+(𝒪_ℂ_p), etc. 

Our first goal is to identify the map  
obtained from π_2 in (<ref>),

    ()^φ = p→^+;

by construction, it is Gal(ℚ_p)-equivariant.
Now we have a (Galois-equivariant) short exact sequence 
    0 →ℚ_p(1) → ()^φ = p→ℂ_p → 0
 as above. 
Furthermore, the map 
(<ref>)
when restricted to the submodule 
ℚ_p(1) ⊂ ( )^φ = p  is
essentially determined: it is given by the
dlog map to derived de Rham cohomology
 as it comes from the usual Chern character ( 𝒪_C;
 ℚ_p) →^-(𝒪_C; ℚ_p), for 𝒪_C. 
 As is proved in <cit.>, the image of a generator of ℚ_p(1) gives a uniformizer of
^+ (which is  a DVR).

Recall that ^+  has a complete, exhaustive
filtration (via powers of the augmentation ideal) with associated graded given
by ℂ_p, ℂ_p(1), ℂ_p(2), …. 
Moreover, there are no Gal(ℚ_p)-equivariant maps
ℂ_p →^+ (see <cit.>).
Thus there is at most one (and hence exactly one, by construction)
Galois-equivariant map
()^φ = p→^+ which extends
the dlog map. 
This shows that the map (<ref>) is actually completely determined by its
behavior on ℚ_p(1). 

Now by a deep result of Fargues–Fontaine, the graded ring ⊕_i ≥ 0(𝒪_C)^φ = p^i is
generated in degree one <cit.>, so the maps for higher i are determined by their
behavior for i  = 1 by multiplicativity. 
In particular, these
observations show that the maps in the fundamental exact sequence, although here
they are produced by topological means, are entirely determined by their value
on ℚ_p(1), as long as they are Galois-equivariant. 
 




 §.§ Application to p-adic nearby cycles


Let C be an algebraically closed,  complete nonarchimedean field of mixed
characteristic (0, p). 
In <cit.>, an explicit description of the ℤ_p(i)
sheaves is given for smooth formal schemes over 𝒪_C. 
Using this, we can recover some cases of comparison results of Colmez–Nizioł 
<cit.> and Tsuji <cit.>, cf. also Kato <cit.>.


 
Let 𝔛 be a formal scheme over 𝒪_C. 
We consider the pro-étale site 𝔛_proet of 𝔛 (equivalently, of its special
fiber), cf. <cit.>. 

For each i, we consider[The functor ψ here should refer to the generic
fiber functor, but we do not define it here to avoid technicalities.] the sheaf of p-adic nearby cycles 
R ψ_* ( ℤ_p(i)), which is a
D(ℤ_p)^≥ 0-valued sheaf
 on 𝔛_proet. 
Explicitly, 
given an affine pro-étale open A →𝔛, 
we have that R Γ( A, R ψ_*(ℤ_p(i))) ≃ 
R Γ_proet(  A[1/p], ℤ_p(i)) is the pro-étale cohomology
of A[1/p] with values in the (usual) sheaf ℤ_p(i).[Here we can consider either the
scheme (A[1/p]) or the rigid analytic generic fiber by the affinoid
comparison theorem <cit.>.] 
 




 

Let R be a formally smooth 𝒪_C-algebra and let 𝔛 =
(R). Then, as sheaves on 𝔛_proet, we have a natural equivalence
ℤ_p(i) ≃τ^≤ i R ψ_*( ℤ_p(i)). 
In particular, it follows that

    ℤ_p(i)(R) ≃ R Γ( 𝔛_proet, 
    τ^≤ i R ψ_*( ℤ_p(i))).

 



To apply this, 
let K be a discretely valued field with perfect residue field k, ring
of integers 𝒪_K ⊂ K, and uniformizer π∈𝒪_K; suppose  
K ⊂ C (e.g., we could take C = K). 
Let 𝔛_0 be a smooth proper formal scheme over 𝒪_K with generic
fiber X_0, a smooth proper rigid space over K. 

[de Rham cohomology of formal schemes and rigid spaces]
We let L Ω_𝔛_0/𝒪_K denote the (p-adic) derived de Rham cohomology of
𝔛_0 over 𝒪_K equipped with its Hodge filtration. 
In fact,  
L Ω_𝔛_0/𝒪_K is also the p-complete usual de Rham complex since
𝔛_0 is formally smooth over 𝒪_K, cf. <cit.>, and the Hodge filtration is a finite filtration. 
    We let (by a slight abuse of notation)
    Ω_X_0/K=(LΩ_𝔛_0/_K)__p denote the rationalization, which we can interpret as
the de Rham cohomology of the rigid generic fiber X_0. 
Note that L Ω_𝔛_0/𝒪_K is a perfect 
𝒪_K-module, and Ω_X_0/K is a perfect K-module. 



We also consider the ring ^+ = ^+(𝒪_C) with its ξ-adic
filtration. 
Together, it follows that 
Ω_X_0/K⊗_K^+ admits a filtration in the
derived ∞-category D(K). 
Then one has the following result, a special case of results of <cit.> in
the case of good reduction; note that <cit.> treat the more general
semistable case, which we do not consider here.  
In the following, all references to , ^+, etc. will implicitly 
be with respect to the perfectoid ring 𝒪_C. 

 
Let 𝔛_0/𝒪_K be a smooth proper formal scheme. 
Let 𝔛 denote the base change of 𝔛_0 to 𝒪_C and
𝔛_0 its reduction modulo π. 
For each i ≥ 0, we have a natural pullback square in D(ℚ_p),

    R Γ( 𝔛_proet, 
    τ^≤ i R ψ_*( ℚ_p(i))) 
    [d]  [r]     
    (⊗_W(k) R Γ_crys(
    𝔛_0/W(k)))^φ = p^i [1/p] [d]  
    Fil^≥ i
    (Ω_X_0/K⊗_K^+)  [r]       (Ω_X_0/K⊗_K^+).

 
 
We claim that this follows from 
<Ref>, applied to 𝔛. 
Note that 
(<ref>) gives a fiber square

    ℚ_p(i)( 𝔛) [d]  [r]    ℚ_p(i)( 𝔛/π) [d] 
    
    ( L Ω_𝔛/𝒪_K^≥ i)_ℚ_p[r]      (
    L Ω_𝔛/𝒪_K)_ℚ_p

in fact, ℚ_p(i)( 𝔛/p) →ℚ_p(i)( 𝔛/π) is
an isomorphism thanks to <Ref>. 


The top left term in (<ref>) is identified via <Ref>. 
For the top right, we observe that there is a natural equivalence

    𝔛⊗_𝒪_C𝒪_C/π≃𝔛_0⊗_k 𝒪_C/π.

For 𝔽_p-algebras, the construction  L WΩ_(-)
satisfies a Künneth formula, so we get

    L WΩ_𝔛/π≃ L W Ω_𝔛_0⊗_W(k).

Note that we do not need to p-complete again,
since 𝔛_0 is smooth and proper. 
Taking Frobenius fixed points, we identify the top right term now rationally,
thanks to (<ref>). 

We can replace the bottom row of (<ref>) with its completion with
respect to the (rationalized) Hodge filtration. 
Recall  that p-adic derived de Rham cohomology together
with its Hodge filtration (so as a filtered 𝔼_∞-algebra)
satisfies a Künneth formula.
Therefore, we have

    L Ω_𝔛/𝒪_K≃ L
    Ω_𝔛_0/𝒪_K⊗_𝒪_K L
    Ω_𝒪_C/𝒪_K,
 
since
the filtration on 
L
Ω_𝔛_0/𝒪_K is finite, and it is by perfect
𝒪_K-modules. 
It follows that the Hodge completion of the rationalization of L
Ω_𝔛/𝒪_K is equivalent,
in the filtered derived ∞-category of K,
to

    Ω_X_0/K⊗_K^+,
 
where we use 
<Ref> for the identification with ^+. 
 













§ TWISTED TATE DIAGONALS
 

In this section we investigate under which conditions Hochschild homology in a general symmetric monoidal ∞-category admits a (twisted) cyclotomic structure.
The main result is Corollary <ref> and the fact that it applies to graded and filtered  as recorded in Examples <ref> and <ref>. 


As usual, we fix a prime p.
Let  be a presentably symmetric monoidal ∞-category and L→ be a symmetric monoidal, left adjoint functor. 


An L-twisted diagonal on  is a symmetric monoidal natural transformation 

    Δ L(C) → (C ⊗…⊗ C)^hC_p

of lax symmetric monoidal functors →. Assume  is
    additionally stable;[In fact,  semiadditive (so that the Tate
    construction is defined) suffices.] then an L-twisted Tate diagonal is a
    symmetric monoidal natural transformation

    Δ L(C) → T_p(C) := (C ⊗…⊗ C)^tC_p .
 





  * The ∞-category of spaces admits a (unique) id-twisted diagonal
 and the ∞-category of spectra admits a (unique) id-twisted
 Tate diagonal, see <cit.>. 

  * More generally, if  admits the cartesian symmetric monoidal structure then it admits a canonical id-twisted diagonal induced by the actual diagonal. 
 




 Suppose 	R is an 𝔼_∞-ring and = Mod_R,
 considered as a symmetric monoidal ∞-category with the R-linear
 tensor product. Then every
 left adjoint, symmetric monoidal functor L is given by an induction along an
 𝔼_∞-map l R → R and we will prove below that the datum of an L-twisted Tate diagonal on  is 
equivalent to an 𝔼_∞-homotopy between the composition
 
    R  R  R^tC_p

 and the Tate-valued Frobenius φ R → R^tC_p of R,
 <cit.>. We shall refer to an 𝔼_∞-ring R with such a datum as a cyclotomic base. An example is R= 𝕊[z], see Example <ref> below. 


 To see that the datum of a twisted Tate diagonal is equivalent to such an
 equivalence, we first note that a symmetric monoidal natural transformation L
 → T_p of functors _R →_R is determined by its restriction to
 the perfect modules _R^ω⊂_R since L preserves filtered colimits. Since T_p is lax symmetric monoidal we get a factorization
 
    _R T_p_T_p(R)res__R

as lax symmetric monoidal functors. Upon restriction to _R^ω the
first functor is given by base-change along the  Tate-valued Frobenius φ R → R^tC_p such that we get a factorization T_p|__R^ω = res_∘ind_φ. Now a symmetric monoidal transformation 

    L|__R^ω = ind_l →res_∘ind_φ

is by adjunction equivalent to a natural transformation

    ind_∘ l = ind_ind_l→ind_φ .

But every object in _R^ω is dualizable and both functors ind_∘ l and ind_φ are symmetric monoidal. Thus every such symmetric monoidal transformation is necessarily an equivalence and thus induced by an equivalence of maps of 𝔼_∞-rings R → R^tC_p. This shows the claim. 



Note that a general  symmetric monoidal, stable ∞-category  does
    not admit L-twisted Tate-diagonals for arbitrary L. For example if we
	 consider = 𝒟(ℤ) ≃Mod_Hℤ
	 then by <Ref> above a twisted Tate diagonal would be the same as a factorization of the Tate-valued Frobenius

    Hℤ→ (Hℤ)^tC_p

through the triv-map Hℤ→ Hℤ^tC_p. These two maps however differ by Steenrod operations as shown in <cit.>. But any twist would be the identity. 


In the following, we use the notation A
to denote the Hochschild homology object of an algebra object A ∈Alg()
internal to ; 
e.g., if =, this recovers . 


Assume that  is equipped with an L-twisted diagonal resp. Tate
    diagonal. Then we get for each algebra object A ∈Alg()
    an induced S^1-equivariant map

    L ( A) → ( A)^hC_p    resp.     L(  A) → ( A)^tC_p

    which is functorial and symmetric monoidal in A.


We closely follow the construction of the cyclotomic structure on  given
in <cit.>. We will mostly indicate the necessary changes and thus recommend that the reader take a look at the construction there first. We treat the case of the Tate diagonal which is the only case that we will need in this paper, but the case of the diagonal works exactly the same. 

We first recall that A is the geometric realization of the cyclic object in  informally written as

    ⋯[r]<1.5pt>[r]<-1.5pt>[r]<4.5pt>[r]<-4.5pt>     A⊗ A⊗ A@(ul,ur)^C_3[r]<3pt>[r][r]<-3pt>      A⊗ A@(ul,ur)^C_2[r]<1.5pt>[r]<-1.5pt>     A .

Thus L(  A) is the geometric realization of the cyclic object 

    ⋯[r]<1.5pt>[r]<-1.5pt>[r]<4.5pt>[r]<-4.5pt>     L(A⊗ A⊗ A)@(ul,ur)^C_3[r]<3pt>[r][r]<-3pt>     L( A⊗ A)@(ul,ur)^C_2[r]<1.5pt>[r]<-1.5pt>     L(A) .

For a given L-twisted Tate diagonal 

    Δ L(C) → (C ⊗ ... ⊗ C)^tC_p  = T_p(C)

we want to construct a natural map of cyclic objects

    ⋯[r]<1.2ex> [r]<-1.2ex>[r]<0.4ex> [r]<-0.4ex>     L(A^⊗ 3)[d]^Δ@(ul,ur)^C_3[r]<0.8ex>[r]<0ex>[r]<-0.8ex> [r]    L(A^⊗ 2) [d]^Δ@(ul,ur)^C_2[r]<0.4ex> [r]<-0.4ex>    L A [d]^Δ
    ⋯[r]<1.2ex> [r]<-1.2ex>[r]<0.4ex> [r]<-0.4ex>    (A^⊗ 3p)^tC_p@(dl,dr)_C_3[r]<0.8ex>[r]<0ex>[r]<-0.8ex> [r]   (A^⊗ 2p)^tC_p@(dl,dr)_C_2[r]<0.4ex> [r]<-0.4ex>   (A^⊗ p)^tC_p,

and obtain the desired map L( A) → ( A)^tC_p as the geometric realization of this map of cyclic objects followed by the canonical interchange map from the realization of the Tate constructions to the Tate construction of the realization. 

In order to construct such a natural transformation of cyclic objects, we
    proceed as in <cit.>: we eventually need to show that we can extend the symmetric monoidal natural transformation Δ L → T_p of functors → to a BC_p-equivariant symmetric monoidal natural transformation of functors  
 from the functor

    L̃ N(Free_C_p)×_N(Fin)^⊗_pr^⊗_⊗L

given by

    (S,(X_s∈S=S/C_p))↦ L(⊗_s∈S X_s)

to the functor

    T̃_p N(Free_C_p)×_N(Fin)^⊗_→ (^⊗_)^BC_p⊗^BC_p-^tC_p

given by

    (S,(X_s∈S=S/C_p))↦ (⊗_s∈ S X_s)^tC_p .

Here Free_C_p is the category of finite free C_p-sets equipped
    with the cocartesian symmetric monoidal structure. The group object
    BC_p-acts on this category in the obvious way and acts trivially on
    . For a precise construction of these functors we refer to
    <cit.>, specifically Proposition III.3.6 and the
    construction around that.

The inclusion

    _⊗(N(Free_C_p)×_N(Fin)^⊗_, ) ⊆_lax(N(Free_C_p)×_N(Fin)^⊗_, )

admits a right adjoint by 
Lemma III.3.3 resp. Remark III.3.5 in <cit.>. Using the same construction and argument as in the proof of 
<cit.> we see that the ∞-category _⊗(N(Free_C_p)×_N(Fin)^⊗_, ) is equivalent to the ∞-category (NTor_C_p, _lax(, )) where Tor_C_p denotes the category of C_p-torsors. 
Under this equivalence the right adjoint to the inclusion is given by restricting a functor 
in _lax(N(Free_C_p)×_N(Fin)^⊗_, ) to NTor_C_p×⊆ N(Free_C_p)×_N(Fin)^⊗_ and forming the adjunct. 

Now the functor L̃ is symmetric monoidal rather than lax symmetric
monoidal. Thus to construct a map from L̃ to T̃_p is by
adjunction equivalent to constructing a transformation in
(NTor_C_p, _lax(, )) between the
respective restrictions. Moreover BC_p-acts on all those categories, i.e., to construct a BC_p-equivariant transformation between L̃ and T̃_p is equivalent to construct a transformation in 

    ^BC_p(NTor_C_p, _lax(, )).

Now the category Tor_C_p is in fact equivalent to BC_p. Since the BC_p-action on _lax(, ) is trivial it follows that the above ∞-category of BC_p-equivariant functors is equivalent to  _lax(, ). 

Taking everything together we see that there is a unique symmetric monoidal transformation L̃→T̃_p extending the transformation Δ L → T_p. 
Together with the constructions above this finishes the proof. 


We shall refer to the map L(  A) → ( A)^tC_p as a twisted cyclotomic structure on A. Thus the last result shows that for ∞-categories with a twisted Tate diagonal we find that Hochschild homology admits a twisted cyclotomic structure.

For a given L-twisted diagonal on , the stabilization () admits a canonical induced (L)-twisted Tate diagonal.


We would like to construct a symmetric monoidal natural transformation

    (L)(C) → ( C ⊗…⊗ C)^tC_p = T_p(C)  .

Such a transformation is by adjunction  the same as a symmetric monoidal transformation

    id→ R' T_p

where R'() →() is the right adjoint to (L).
We now use that the functor Ω^∞ induces an equivalence

    _lax^Ex((), ()) →_lax^Ex((), )

by <cit.>. It follows that it suffices to  construct a symmetric monoidal transformation

    Ω^∞→Ω^∞ R' T_p

of functors () →. We denote by R→ the right adjoint to the functor L→. Then we have an equivalence Ω^∞ R' ≃ R Ω^∞ of lax symmetric monoidal functors which follows from the fact that the left adjoint diagram

    [r]^L[d]^Σ^∞   [d]^Σ^∞
    () [r]^(L)   ()  .

commutes (up to symmetric monoidal equivalence). As a result we need to construct a symmetric monoidal natural transformation

    Ω^∞→ R Ω^∞ T_p  .

Now we use that we have  canonical symmetric monoidal transformations

    γ (Ω^∞ C ⊗…⊗Ω^∞ C)^hC_p→Ω^∞( (C ⊗…⊗ C)^hC_p)
    →Ω^∞( (C ⊗…⊗ C)^tC_p)

where the first one is induced by the lax symmetric monoidal structure of Ω^∞ together with the fact that it commutes with limits and the second by the canonical map from homotopy fixed points to the Tate construction.

Now we use the unstable diagonal on  to get as the adjoint a symmetric monoidal natural transformation

    Ω^∞ C → R (Ω^∞ C ⊗…⊗Ω^∞ C)^hC_p

and compose it with the map R(γ) above to get a symmetric monoidal
    natural transformation as in (<ref>).


For every symmetric monoidal ∞-category I we consider the symmetric monoidal functor l_p I → I given by sending i to i^⊗ p. We let 

    L_p(I, 𝒮) →(I,𝒮)

be left Kan extension along l_p. We equip the category (I, 𝒮) with the
Day convolution symmetric monoidal structure. Then the left Kan extension L_p becomes symmetric monoidal.

Assume that the ∞-category I has the following property:
for every pair of objects i,j ∈ I we have that the canonical forgetful map

    Map_I(i^⊗ p, j)^hC_p→Map_I(i^⊗ p, j)

is an equivalence of spaces.[Note that an equivalent way of stating this condition is to say that the homotopy orbits (i^⊗ p)_hC_p exist in I and the map i^⊗ p→ (i^⊗ p)_hC_p is an equivalence. ]
Then the inverse of the map (<ref>) induces a canonical L_p-twisted diagonal on (I, 𝒮). 


We consider the symmetric monoidal (co)Yoneda embedding 

    I^→(I, 𝒮)  .

Then symmetric monoidal transformations  

    L_p(C) → (C ⊗…⊗ C)^hC_p

as functors (I, 𝒮) →(I, 𝒮)
are the same as symmetric monoidal transformations between the restrictions of the functors along the Yoneda embedding. The restricted functors I^→(I, 𝒮) are given by the lax symmetric monoidal assignments

    i ↦ (j ↦_I(i^⊗ p, j))     and    
        i ↦ (j ↦_I(i^⊗ p, j)^hC_p)  .

The canonical map _I(i^⊗ p, j)^hC_p→_I(i^⊗ p,
    j) is a symmetric monoidal natural transformation. By assumption it is an equivalence so that the inverse induces the required transformation.




For a general symmetric monoidal ∞-category I the category (I,
𝒮) does not admit an L_p-twisted diagonal. As an example consider
any  cocartesian symmetric monoidal ∞-category I. Then the Day
convolution structure on (I, ) is cartesian.[This follows from
the fact that generally Day convolution for a cocartesian source is given by the
pointwise tensor product, which in our case happens to agree with the cartesian
product.] Thus an L_p-twisted diagonal would amount to a natural symmetric monoidal transformation

    F^× p→ (F^× p)^hC_p

which does not exist. 

But note that this category admits an id-twisted diagonal. This raises the question if for every symmetric monoidal ∞-category I there is a twist on (I, 𝒮) and a twisted diagonal. The answer to this question is also `no' in general but we will not go into the intricacies of concrete counterexamples here.
 


If I is a symmetric monoidal ∞-category satisfying the condition of Lemma <ref> then we have for every algebra A in (I, ) a twisted cyclotomic structure on A, i.e.  an S^1-equivariant map

    L_p( A) → ( A)^tC_p .

This map is natural and symmetric monoidal in A.


Combine Proposition <ref> with Lemma <ref> and Lemma <ref>.





We consider the category I = ℤ_≥ 0^ds. Then 
(I, Sp) is the ∞-category of graded spectra. The category
I obviously satisfies the condition of Lemma <ref>. Thus we get
that for a graded ring R_∙, graded  admits an L_p-twisted cyclotomic structure or equivalently a sequence of S^1-equivariant maps 

    (R)_i →(R)_pi^tC_p .

The same logic applies to spectra graded over any discrete monoid in place of ℤ_≥ 0^ds.



Consider the ∞-category I = ℤ_≥ 0^ associated to the poset of positive integers. Then this also satisfies the condition of Lemma <ref>. The category of functors (I, ) is given by filtered spectra and thus filtered  of a filtered ring spectrum R admits a filtered cyclotomic structure, i.e. S^1-equivariant maps

    Fil^≥ i(R) →  (Fil^≥ pi(R))^tC_p.




Consider the category I = B ℤ_≥ 0. This category also obviously satisfies the condition of Lemma <ref>. Thus the category

    (I, ) ≃Mod_𝕊[z]

admits a twisted Tate diagonal and thus relative  admits a (twisted)
cyclotomic structure, as is used in <cit.>. The twist L_p corresponds to the map l𝕊[z] →𝕊[z] sending z to z^p. 


We want to end this section by remarking some functorialities of the twisted cyclotomic structures.

A symmetric monoidal category with (Tate) diagonals consists of a triple (, L, Δ) as in Definition <ref>. A map of symmetric monoidal categories with (Tate) diagonals 

    (, L, Δ) → (', L', Δ')

is given by a left adjoint symmetric monoidal functor F→' together with a symmetric monoidal equivalence L' ∘ F ≃ F ∘ L and a natural symmetric monoidal equivalence between the two maps

    L'(FX) → (FX⊗…⊗ FX)^tC_p

induced from Δ and Δ' (both sides considered as lax symmetric monoidal functors →').


Form the construction of the twisted cyclotomic structure in Proposition <ref> we see immediately that for such a map of symmetric monoidal ∞-categories with Tate diagonals we get an equivalence of twisted cyclotomic objects

    F(  A ) ≃(FA)

for every algebra A in . Here the first object F( A) is twisted cyclotomic by the composition

    LF(  A) ≃ F L (  A) F φ F(  A^tC_p) → F(  A)^tC_p .

We also have a relative analogue of Lemma <ref>: every map of
symmetric monoidal ∞-categories with diagonals induces upon
stabilization a map of symmetric monoidal ∞-categories with Tate
diagonals. This is straightforward to prove. Finally there is also an analogue
of Lemma <ref> which we  will state and prove now.


Assume that f I → I' is a symmetric monoidal functor such that I and I' satisfy the condition of Lemma <ref>. Then left Kan extension along f induces a map of symmetric monoidal ∞-categories with diagonals

    ((I, 𝒮), L_p, Δ) → ((I', 𝒮), L'_p, Δ')  .

where L_p, L_p', Δ and Δ' are as in Lemma <ref>.


We have a commutative square

    I [r]^f[d]^l_p    I'[d]^l_p'
    
    I [r]^f     I'

for the functors l_p(i) = i^⊗ p and l_p'(j) = j^⊗ p. Thus we get an induced square of the left Kan extensions

    (I, 𝒮) [r]^F[d]^L_p   (I', 𝒮)[d]^L_p'
    (I, 𝒮) [r]^F    (I', 𝒮)  .

This provides the first part of the datum of a map of symmetric monoidal ∞-categories with Tate diagonals. We now also have to provide an equivalence of two different natural transformations between two functors

    (I, 𝒮) →(I', 𝒮)  .

Such a transformation is determined by its restriction to I^⊆(I, 𝒮) and there the functors are given by

    i ↦(j ↦Map_I'(f(i)^⊗ p, j))

and 

    i ↦(j ↦Map_I'(f(i)^⊗ p, j)^hC_p)  .

Unravelling the constructions we see that both of the two transformations are given by the inverse of the canonical forgetful map 

    Map_I'(f(i)^⊗ p, j)^hC_p→Map_I'(f(i)^⊗ p, j)

and thus are canonically equivalent.


From these statements together we can deduce the following corollary:


Assume that f I → I' is a symmetric monoidal functor such that I and I' satisfy the condition of Lemma <ref>. Then for every algebra A ∈(I, Sp) we have an equivalence of L'_p twisted cyclotomic objects 

    F(  A ) ≃(FA)

where F is left Kan extension along f.



For a graded ring spectrum R_∙ we have that the direct sum 

    ⊕_i(R_∙)_i

is equivalent to (⊕_i R) as cyclotomic spectra. Similarly, for a filtered ring spectrum R we have that the filtered cyclotomic structure refines the cyclotomic structure on (R). 


We finally note that one can also look at the functor 

    ev_0(ℤ_≥ 0^ds, ) →

given by restriction to the 0-th component. We claim that this also refines
    to a map of symmetric monoidal ∞-categories with Tate diagonals.
    This can be seen by verifying the corresponding unstable statement which is
    straightforward  using an argument similar to the one in the proof of
    Corollary <ref>. This then shows that the cyclotomic structure
    on the 0-th graded component (R_∙)_0 agrees with the one on
    (R_0) for every graded ring spectrum R_∙.



§ CATEGORICAL LEMMAS
 
 
[Left Kan extensions] 

Let R be a ring, and 
let Poly_R  be the category of finitely generated polynomial R-algebras. Given a presentable ∞-category
𝒞 and an accessible functor fPoly_R →𝒞, we can left Kan
extend to obtain a functor LfSCR_R →𝒞 which commutes
with geometric realizations, for
SCR_R the ∞-category of simplicial commutative R-algebras. 
Compare <cit.> and <cit.>. 
 

Let (ℒ, ℛ) 𝒞⇄𝒟 be an adjunction 
of ∞-categories. Then 
for any ∞-category ℰ, we obtain an adjunction

    (ℛ^*, ℒ^*) = (f ↦ f ∘ℛ, f' ↦ f ∘ℒ)( 𝒞, ℰ) ⇄(𝒟,
    ℰ)  .



 

Let f_1, f_2𝒟→ℰ be functors. 
Suppose that for any x ∈𝒟, the natural map f_1( ℒℛ x ) → f_1(x) is an equivalence. 
Then 
we find

    _(𝒟, ℰ)(f_1, f_2) 
    ≃_(𝒞, ℰ)(f_1 ∘ℒ, f_2 ∘ℒ).

This follows from the adjunction (<ref>). 
 



Now we specialize to the case where 𝒞 = is the
∞-category of simplicial commutative rings and 𝒟 =
_𝔽_p is the ∞-category of simplicial commutative
𝔽_p-algebras. 
We have an adjunction (ℒ, ℛ)⇄_𝔽_p, where the
left adjoint is R ↦ R ⊗^L_ℤ𝔽_p and the right adjoint is
simply the forgetful functor. 

For any R ∈_𝔽_p, we have a canonical endomorphism φ
R → R, the Frobenius. 

 

Let R ∈_𝔽_p. 
There is a natural map f R → R ⊗^L_ℤ𝔽_p in 
_𝔽_p
such
that the composites 
R f→ R ⊗^L_ℤ𝔽_p → R and
R ⊗^L_ℤ𝔽_p → R → R ⊗^L_ℤ𝔽_p are the respective Frobenius endomorphisms. 
 
 
It suffices to assume that R is discrete (even a finitely generated
polynomial ring) via left Kan extension. 
In this case, R ⊗^L_ℤ𝔽_p is concentrated in
homological degrees zero  and one (with π_0 = R itself), and one knows that the Frobenius endomorphism 
annihilates π_1, cf. <cit.>. Thus, the Frobenius map
R⊗_^L_p→ R⊗_^L_p factors canonically through the truncation map
R⊗_^L_p→π_0(R⊗_^L_p) R.
This gives the map f as desired.
 

 

Let F_1, F_2_𝔽_p→ D(ℤ) be two functors. 
Suppose F_1 has the property that the natural map F_1(R) → F_1(R) given
by Frobenius is multiplication by p^i. 
Then for any natural transformation 
u F_1( - ⊗^L_ℤ𝔽_p) → F_2(-
⊗^L_ℤ𝔽_p) of functors _ℤ→ D(ℤ), we have that p^i u arises from a
natural transformation 
F_1 → F_2.
In fact, we have 

    _(_, D(ℤ))( F_1( - ⊗_ℤ_p^L
    𝔽_p), F_2( - ⊗_ℤ_p^L 𝔽_p)) [1/p]
    ≃_(_𝔽_p, D(ℤ))( F_1, F_2) [1/p].

 
 
This follows from (<ref>) in the case 
of the adjunction (ℒ, ℛ) _⇄_𝔽_p. 
By construction, we are given a map 
ℒ^* F_1 →ℒ^* F_2 of functors _𝔽_p→ D(ℤ), or equivalently by adjointness a map 
ℛ^*ℒ^* F_1 → F_2 of functors _→ D(ℤ). 
Now we have  a natural map F_1 →ℛ^* ℒ^* F_1 
given by the natural map 
f  R → R ⊗^L _ℤ_p𝔽_p of <Ref>;
it has the property that the composites in either order with the adjunction
map ℛ^* ℒ^* F_1 → F_1 are given by multiplication by
p^i. 
The composition
F_1→ℛ^* ℒ^* F_1 → F_2 defines the desired map F_1 → F_2. 
This argument also proves the displayed equation. 
 



Let K be a complete discretely valued field with ring of integers
𝒪_K ⊂ K and residue 
field k; let π∈𝒪_K be a uniformizer. 
Let _𝒪_K denote the category of topologically finitely generated, formally smooth
𝒪_K-algebras and let _k denote the category of smooth
k-algebras. 

We now give a similar result for functors defined on a restricted class of
simplicial commutative k-algebras.
For the next result, we will argue similarly, but with a smaller set of ∞-categories. 
For these finiteness conditions, see
<cit.> (in the slightly more complicated 𝔼_∞-case). 
 


    
  (1)
Let ^_k denote the ∞-category of simplicial commutative
k-algebras 
R
which are almost finitely presented: equivalently, 
π_0(R) is finitely generated as a k-algebra and each π_i(R) is a
finitely generated π_0(R)-module. 
Equivalently, R belongs to ^_k if and only if R can be written as
the geometric realization of a simplicial diagram of finitely generated
polynomial k-algebras. 

  (2)
Similarly, we define ^_𝒪_K to be the ∞-category of 
π-complete 
simplicial commutative
𝒪_K-algebras R such that 
π_0(R) is topologically finitely generated 
over 𝒪_K (i.e., a quotient of a π-completed polynomial ring)
and each  π_i(R) is finitely generated over R. 
Equivalently, R belongs to ^_𝒪_K if and only if R can be written as
the geometric realization of a simplicial diagram of π-completed finitely generated
polynomial 𝒪_K-algebras. 
Yet another characterization is that R should be almost finitely presented
over the π-completion of a finitely generated polynomial algebra over
𝒪_K with a structure map that is surjective on π_0. 

 
 

Let ℰ be an ∞-category admitting sifted colimits. 
Let F_1, F_2_k^afp→ℰ be functors. If

    
  (1) F_1  commutes with geometric realizations and
    
  (2)
	 F_1(R) ≃ F_1( π_0 R) for R ∈_k^afp,

then 

    _(_k, ℰ)(F_1, F_2) 
    ≃_( _𝒪_K, ℰ)( F_1( -
    ⊗_𝒪_K k),
    F_2(- ⊗_𝒪_K k)).

 

 
Since F_1 is left Kan extended from smooth (even finite type polynomial) k-algebras
as it commutes with geometric realizations, 
we have 

    _(_k, ℰ)(F_1, F_2) 
     ≃_( _k^, ℰ)(F_1, F_2) .

Similarly, 

    _(_𝒪_K, ℰ)(F_1( -
    ⊗_𝒪_K k), F_2( - ⊗_𝒪_K k)) 
     ≃_( _𝒪_K^,
     ℰ)(F_1(- ⊗_𝒪_K k), F_2(- ⊗_𝒪_K k)) ,

because F_1(- ⊗_𝒪_K k)_𝒪_K^→ℰ  is left Kan extended from
_𝒪_K. Now we have an adjunction 
_𝒪_K^⇄_k^ given by
base-change and restriction of scalars. Thus, the result follows as in
<Ref>. 
 






amsalpha




