Total $p$-differentials on schemes over $Z/p^2$
Taylor Dupuy Eric Katz Joseph Rabinoff David Zureick-Brown
math.AG math.NT



For a scheme X defined over the length 2 p-typical Witt vectors W_2() of a characteristic p field, we introduce total p-differentials which interpolate between Frobenius-twisted differentials and Buium's p-differentials. They form a sheaf over the reduction X_0, and behave as if they were the sheaf of differentials of X over a deeper base below W_2(). This allows us to construct the analogues of Gauss–Manin connections and Kodaira–Spencer classes as in the Katz–Oda formalism.  We make connections to Frobenius lifts, Borger–Weiland's biring formalism, and Deligne–Illusie classes.





Kaonic deuterium and low-energy antikaon-nucleon interaction
    Wataru Horiuchi1whoriuchi@nucl.sci.hokudai.ac.jp Tetsuo Hyodo2hyodo@yukawa.kyoto-u.ac.jp Wolfram Weise3weise@tum.de
        

    December 30, 2023
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§ INTRODUCTION


In a series of papers, Alexandru Buium introduced a notion of arithmetic derivations, extending the analogy between function fields like [[t]] and mixed characteristic local fields like _p^  (see the textbook <cit.>). For example, there is an arithmetic derivation operator on _p^, called a p-derivation, defined by

    δ x=ϕ(x)-x^p/p,

where ϕ is the Frobenius. While δ does not obey the usual Leibniz rule and is not even additive, it has many formal properties in common with derivations. Just as the usual notion of derivation measures non-constancy of an object over a base, this arithmetic derivation measures how much the Frobenius action on an object over _p^ differs from x↦ x^p:  the arithmetic analogue of triviality of a family is the existence of a lift of Frobenius. This is consistent with Borger's approach to the field of one element <cit.>, where one views lifts of Frobenius as a necessary condition for descent to _1, just as a vector field on a family extending that on the base is a necessary condition for descent from [[t]] to .


More generally, Buium <cit.> classified all arithmetic analogues of derivations on a local ring R of characteristic zero.  By an “arithmetic analogue of a derivation,” we mean an operation θ R → R on a ring R, which has a sum rule, a product rule, and which is appropriately normalized to take certain values on 0 and 1. 
Buium proved that for R a local ring of characteristic zero, there can only exist four types of operations up to equivalence: derivations, difference operations, p-derivations, and p-difference operations where p is some prime. These various notions afford analogous definitions of differentials and jet spaces.

It turns out that operations θ satisfying the conditions imposed  in <cit.> are intimately tied to ring schemes. 
Under this correspondence, usual derivations correspond to the ring scheme of dual numbers, and Buium's p-derivations correspond to the ring of p-typical Witt vectors of length two. 

In fact, the ring scheme formalism of Borger–Weiland <cit.> provides a uniform framework for formulating differentials and  jet spaces as well as working over a base.







One classical construction using differentials that has been missing its p-differential analogue is that of taking differentials on the total space. Specifically, if one has a scheme π X→ S smooth over a base, one may form the exact sequence

    0[r]   π^*Ω^1_S [r]   Ω^1_X [r]   Ω^1_X/S[r]   0.

This construction is crucial to Katz–Oda's construction of the Gauss–Manin connection <cit.>. The purpose of this paper is to construct an analogue of the restriction of this exact sequence to a closed fiber, at least for the case where the base is /p^2. In fact, we will consider schemes defined over /p^2 (or more generally W_2()) as families over a “deeper” base below /p^2 and measure their non-triviality. In this sense, our work comes into contact with the philosophy of the field of one element. The geometric counterpart of our construction can be found in <cit.>. 

This paper accomplishes the goal by introducing a new class of arithmetic derivations from a ring modulo p^2 to a ring modulo p that interpolates between Buium's p-derivations and ordinary Frobenius semi-linear derivations.  We call these total p-derivations. They globalize on schemes X defined over W_2(), the p-typical Witt vectors length 2 of a field of characteristic p. In fact, their dual notion, total p-differentials form a sheaf Ω_X^1, on X_0, the reduction of X. They fit into an exact sequence

    0[r]    _X_0[r]^>>>>>α   Ω_X^1,[r]^>>>>>β   F_X_0^*Ω^1_X_0[r]   0
 
where F_X_0^*Ω^1_X_0 is the pullback by absolute Frobenius of the sheaf of differentials on X_0. The appearance of _X_0 is a reflection of the fact that W_2() should be thought of as an object with a one-dimensional cotangent space over a “deeper” scheme.

Regarding p-differentials as analogous to differentials on a total space, we construct analogues of the Gauss–Manin connection and the Kodaira–Spencer class.  The usual Gauss–Manin connection is a measure of the non-triviality of a local system arising as the cohomology of a family; our analogue measures the existence of lifts of Frobenius.  This gives an alternative view of Deligne–Illusie's deformation-theoretic approach <cit.> to studying lifts of Frobenius.
The arithmetic analogue of the Kodaira–Spencer class via a different construction already appears with applications to diophantine geometry in <cit.>. 
We also give a description of total p-differentials in terms of the biring formalism. Further discussion of the  arithmetic analogue of Kodaira–Spencer classes is in <cit.>.









§ TOTAL P-DIFFERENTIALS




 §.§ Total p-derivations


Let  be a field of characteristic p>2. Let R=W_2() be the p-typical Witt vectors <cit.> of length 2. Let ϕ R→ R be the Frobenius homomorphism. 
Let A be a finitely generated flat R-algebra with mod p reduction A_0. For r∈ R, we will write r for the element r· 1∈ A.
Let C_p(X,Y) be the polynomial

    C_p(X,Y)=X^p+Y^p-(X+Y)^p/p.


 For an A_0-module D_0, a total p-derivation of A into D_0 is a map δ A→ D_0 such that


  * for a,b∈ A, δ(a+b)=δ a+δ b+(C_p(a,b))δ p,

  * for a,b∈ A δ(ab)=(δ a)b^p+a^p(δ b), and

  *  for r∈ R, δ r=((ϕ(r)-r^p)/p) (δ p).


Here, we treat D_0 as an A-module via the natural quotient A→ A_0. If δ p=0, then the derivation is a Frobenius semi-linear derivation. If δ p=1, then it is a p-derivation. Both of these notions are central to <cit.>. Total p-derivations interpolate between them.



For each c∈/p we may specify a total p-derivation θ_p,c/p^2 →/p by setting (following item <ref>):

    θ_p,c(r)=c (r-r^p/p).

So, θ_p,c(p)=c. 
Observe that  total p-derivations /p^2 →/p are determined by their value on p: 
if θ and θ' are two such derivations then θ-θ' is a Frobenius semi-linear derivation, and the only Frobenius semi-linear derivation from /p^2 to /p is zero. 





 §.§ Witt interpolation


Another view of total p-derivations can be given in terms of a construction interpolating between length 2 Witt vectors and a twisted infinitesimal thickening. Let D_0 be a -algebra. For c∈ D_0, let U_c(D_0) be the ring whose underlying set is D_0× D_0 equipped with the operations

    (x_0,x_1)+(y_0,y_1)   =   (x_0+y_0,x_1+y_1+cC_p(x_0,y_0))
    
    (x_0,x_1)· (y_0,y_1)   =   (x_0y_0,x_0^py_1+y_0^px_1).

It is easily seen that U_1(D_0)=W_2(D_0). On the other hand, U_0(D_0) is an infinitesimal thickening of D_0 twisted by Frobenius. The ring U_c(D_0) can be equipped with an R-algebra structure (using the fact that (1,0) is the multiplicative identity) by

    r· (1,0)=(r,c(ϕ(r)-r^p/p)).

Observe that p· (1,0)=(0,c).
For any value of c, there is a homomorphism U_c(B_0)→ B_0 given by (x_0,x_1)↦ x_0. The kernel I satisfies I^2=0 and pI=0.

The proof of the following is straightforward.

For e∈ A_0, there is a homomorphism h_e U_c(D_0)→ U_ce(D_0) given by h(x_0,x_1)=(x_0,ex_1).


 
Given a homomorphism f A_0→ D_0 making D_0 into an A_0-module, total p-derivations δ of A into D_0 with δ p=c∈ D_0 are in one-to-one correspondence with R-algebra homomorphisms

    (f,δ) A→ U_c(D_0),    a↦ (f(π_0(a)),δ a)

where π_0 A→ A_0 is reduction.


 Let g A→ B be formally smooth. Let D_0 be a B_0-algebra, with a A_0-algebra structure induced by g.
Then a total p-derivation δ_A of A into D_0 has a lift δ_B of B into D_0 (that is δ_A=δ_B∘ g). If, in addition, g is formally unramified, this lift is unique.



Let c=δ_A(p). Then, the total p-derivativation δ_A induces a homomorphism (f,δ_A) A→ U_c(D_0). Consider the commutative diagram

    B[r]^h@–>[dr]   D_0
    
    A[u]^g[r]   U_c(D_0).[u]

The top horizontal arrow h is the B_0-algebra structure, and the right vertical arrow is reduction. The dotted arrow exists by formal smoothness (and is unique if f is formally unramified). It must be of the form (h,δ_B) for a total p-derivation δ_B.


The construction of U_c is functorial: if h B_0→ C_0 is a A_0-algebra homomorphism, then there is an induced homomorphism h U_c(B_0)→ U_h(c)(C_0). 



 §.§ Total p-differentials


The module of total p-differentials Ω^1,_A is the A_0-module generated by the symbols d^x for x∈ A subject to the relations


  * for a,b∈ A, d^(a+b)=d^a+d^b+(C_p(a,b))d^p,

  * for a,b∈ A, d^(ab)=(d^a)b^p+a^p(d^b), and

  * for r∈ R,  d^r=((ϕ(r)-r^p)/p) (d^ p).

Here, we treat Ω^1,_A as an A-module with the A-action factoring through the reduction A→ A_0
There is a natural total p-derivation δ A→Ω^1,_A that satisfies the following universality property: for any total p-derivation δ' A→ D_0, there is a unique A_0-module homomorphism gΩ_A^1,→ D_0 such that δ'=g∘δ. 

  For A=R[x_1,…,x_n], a free polynomial algebra, Ω_A^1, is a free A_0-module of rank n+1, generated by d^ p,d^x_1,…,d^x_n.










There are pullbacks for total p-differentials: given an R-algebra homomorphism f A→ B, there is a well-defined homomorphism of B_0-modules, 

    Ω_A^1,⊗ B_0→Ω_B^1,

taking d^a⊗ b_0↦ b_0d^(f(a)).

For an A_0-module M_0, we will consider F_A_0^*M to be the Frobenius tensor product of M: F_A_0^*M_0 is M_0⊗_A_0 A_0 where A_0 is given an A_0-action twisted by Frobenius (for r∈ A_0, x∈ A_0, r· x=r^px.)
The quotient of Ω_A^1, by d^p is F_A_0^*Ω^1_A_0.
There is a natural A_0-module homomorphism α A_0→Ω_A^1, given by α(x)=x d^ p. 
 A left-inverse of α is a homomorphism hΩ_A^1,→ A_0 with h(d^p)=1. 
 
We will relate left-inverses of α to lifts of the absolute Frobenius. Here, a lift of absolute Frobenius is a ring homomorphism ϕ A→ A such that ϕ restricts to the Frobenius on R and makes the following diagram commute

    A[r]^ϕ[d]_π_0   A[d]^π_0
    
    A_0[r]^x↦ x^p    A_0



In much of the literature, one lifts the relative Frobenius which is a -linear homomorphism A_0→ A_0^(p). Here, A_0^(p)=A_0⊗_ (with the -algebra structure on  being given by r· x=r^p x) so for c∈, cx⊗ 1=x⊗ c^p. Lifts of relative Frobenius are in bijective correspondence with lifts of absolute Frobenius. Indeed, one has a commutative diagram

    A_0   [l]_>>>>>>F A_0^(p)   [l]_>>>>>G@/_1.5pc/[ll]_F_A_0 A_0
       [u]   [l]_x^p x[u]

where F is the relative Frobenius given by x⊗ c↦ cx^p and G is the structure homomorphism given by x↦ x⊗ 1. Similarly, we may form A^(ϕ)=A⊗_R R where R is given an R-algebra structure by r· x=ϕ(r)x. A lift of relative Frobenius is an R-homomorphism Φ A→ A^(ϕ) reducing to F.  Throughout this paper, except in remarks, we will use absolute Frobenius which will be denoted by subscripted F. 



  Left-inverses hΩ_A^1,→ A_0 of α are in bijective correspondence with lifts of Frobenius on A where the correspondence is given by

    h↦ [ϕ_h x↦ x^p+ph(d^x)].




Let h be a splitting. Define ϕ_h A→ A by ϕ_h(x)=x^p+ph(d^x). We observe that

    ϕ_h(x+y)   =   (x+y)^p+ph(d^(x+y))
       =   (x+y)^p+ph(d^x+d^y+C_p(x,y)d^p)
       =   x^p+y^p+ph(d^x+d^y)
       =   ϕ_h(x)+ϕ_h(y)

and that

    ϕ_h(xy)   =   (xy)^p+ph(d^(xy))
       =   (xy)^p+ph(x^p d^y+y^p d^x)
       =   (x^p+ph(d^x))(y^p+ph(d^y))
       =   ϕ_h(x)ϕ_h(y).


If ϕ A→ A is a lift of Frobenius, then we define h A→ A_0 by h(x)=(ϕ(x)-x^p)/p (which is well-defined because A is a flat R-algebra). We claim that h factors through d^ A→Ω^1,. This follows from reversing the above arguments and noting that ϕ restricts to the usual Frobenius on R. 

  Let A be a smooth R-algebra. Then, 
there is an exact sequence of A_0-modules,

    0[r]   A_0[r]^>>>>>α   Ω_A^1,[r]^>>>>>β   F_A_0^*Ω^1_A_0[r]   0.




We only need to show that α is an injection. Because we are in the affine setting, there is a lift of Frobenius on A by smoothness, and therefore a left-inverse of α.




 §.§ Globalization


The above constructions globalize by gluing. 
Recall that on an open affine U_0= A_0⊂ X_0, if the quasicoherent sheaf   corresponds to a module M, the pullback by absolute Frobenius F_X_0^* on U corresponds to F_A_0^*M.
Consequently, for X smooth over R, the short exact sequence of Lemma <ref> globalizes to a sequence of sheaves of _X_0-modules, the fundamental exact sequence of total p-differentials:

    0[r]    _X_0[r]^>>>>>α   Ω_X^1,[r]^β   F_X_0^*Ω^1_X_0[r]   0

This exact sequence is an imprecise analogue of the first exact sequence of Kähler differentials <cit.>, albeit with the differentials twisted by Frobenius. If we were to imagine that there is a sequence of projections X→ R →: the non-existent “Ω^1_R/" would be the sheaf R_0 which pulls back to _X_0.

The existence of the locally split short exact sequence immediately yields the following.
 If X is smooth over R, then Ω_X^1, is locally free.


Given a morphism f X→ Y over R, there is a morphism of sheaves on X_0, f^*Ω^1,_Y→Ω^1,_X.

  Let f X→ Y be a smooth morphism of smooth schemes over R. Then f induces an injection f^*Ω_Y^1,→Ω_X^1,. If, in addition, f is étale, then f is an isomorphism.




We have a commutative diagram of locally free sheaves with exact rows:

    0[r]    _X_0[r]^α   Ω_X^1,[r]^β   F_X_0^*Ω^1_X_0[r]   0
    
    0[r]    f^*_Y_0[r]^α[u]   f^*Ω_Y^1,[r]^β[u]   f^*F_Y_0^*Ω^1_Y_0[r][u]   0.

The left vertical arrow is an isomorphism while the right vertical arrow is injective (and an isomorphism if f is étale).


By globalizing Lemma <ref>, we obtain the following:

 Splittings of the fundamental exact sequence (<ref>) are in bijective correspondence with lifts of Frobenius. The set of lifts of Frobenius is a torsor over H^0(X_0,(F_X_0^*Ω^1_X_0,_X_0)).



A splitting of the fundamental exact sequence locally gives lifts of Frobenius which glue into a global lift.
Any two splittings differ by a global section of (F_X_0^*Ω^1_X_0,_X_0). 



 §.§ Total jet spaces


Total p-differentials were originally developed by the authors to recontextualize some arguments of Buium <cit.>. Specifically, given a scheme X over W(), Buium introduced the p-jet space X^1. Its reduction to , X_0^1 is a torsor for the Frobenius tangent space (that is, the pullback of the tangent space of X_0 by absolute Frobenius) and is related to the Greenberg transform. In <cit.>, Buium considers the projective completion of X_0^1 (considered as an affine torsor). By making use of the fundamental exact sequence,


one can see that the projectivization of Ω_X^1, is manifestly a projective completion of the dual to F_X_0^*Ω^1_X_0. This gives a useful functorial understanding of such completions.











 §.§ Birings


In  <cit.>, Borger and Weiland make use of the formalism of birings, which are ring objects in the category of rings, to put certain universal constructions in algebra on equal footing. One has two subcategories of rings _1,_2 and wishes to study functors F_1→_2 between them. The example to keep in mind is the formation of Witt vectors from the category of /p-algebras to the category of /p^2-algebras. In certain cases, the functor will be representable by a biring. That is, there is a ring Q equipped with the data of coaddition Δ^+ Q→ Q⊗ Q and comultiplication Δ^× Q→ Q⊗ Q along with additive and multiplicative counits and additive antipodes such that for A∈_1,

    F(A)=(Q,A)

where  denotes ring homomorphisms. The coaddition and comultiplication induce addition and multiplication on F(A).
One may further consider categories _i of algebras over fields _i and then incorporate the data for algebra structures.

We will describe the construction in terms of /p and /p^2 knowing that they can be replaced by  and W_2() for a characteristic p field k. For c∈/p, we will consider the Witt interpolation U_c as a functor from /p-algebras to /p^2-algebras.
We define 

    Q_c = (/p)[e,η]

with the rules 
 
    Δ^+(e)   =    e⊗ 1+1⊗ e
    Δ^+(η)    =   η⊗ 1 + 1 ⊗η - c ∑_j=1^p-11/pp  j e^p-j⊗ e^j 
    Δ^×(e)   =   e⊗ e
    Δ^×(η)    =    e^p⊗η + η⊗ e^p

 with the other structures defined naturally. It is straightforward verification to show the following:
 
 For any k=/p-algebra D_0, there is an isomorphism:
 
    _/p(Q_c,D_0)   →   U_c(D_0)
    
    f   ↦   (x_0,x_1)=(f(e),f(η)).



The image of f under this map (in the case c=1) is easily seen to be the first two Witt components. Furthermore, we may implement the /p^2-algebra structure on U_c(D_0) by the map

    β/p^2   →   _Z/p^2(Q,D_0)
    
    a   ↦   (a,θ_c(a))

(where θ_c/p^2→/p is the unique total p-derivation with θ_c(p)=c)
so that for e∈/p^2, e· x=(β(e))x where the product is taken in U_c(D_0).
 
There is an enriched tensor product operation ⊙ that acts as an left adjoint to Homs of algebras so that if D_0 is a /p-algebra and B_1 is a /p^2-algebra,

    _/p(Q_c⊙ B_1,D_0)=_/p^2(B_ 1,_/p(Q_c,D_0)).

Total p-differentials can be constructed in this formalism. One defines Q_t=(/p)[t,e,η] using the above biring structure with the indeterminate t substituted for c. Then, for a /p^2-algebra B_1, Q_c⊙ B_1 is the free symmetric (B_1⊗_/p^2/p)-algebra on the total p-differentials Ω^1,_B_1.
 

































§ EXTENSION CLASSES FROM THE FUNDAMENTAL EXACT SEQUENCE


In this section, we will suppose that X is smooth over R. Recall that a splitting of the fundamental exact sequence, if it exists, induces a lift of Frobenius on X.  As an extension, the fundamental exact sequence gives an element

    κ∈^1(F_X_0^*Ω^1_X_0,_X_0)=H^1(X_0,F_X_0^*TX_0)

 (where TX_0 is dual to Ω^1_X_0) that is the obstruction to a lift of Frobenius. 
We might call this the arithmetic Kodaira–Spencer class. If we imagine R to be a point with a vector in the “p-direction”, this would be analogous to the Kodaira–Spencer map applied to this vector. The extension class is interpreted as a Čech class by picking a covering U_i of X such that there is a splitting σ_i F_(U_i)_0^*Ω^1_(U_i)_0→Ω^1,_U_i. Then,  s_ij=σ_i-σ_j gives a homomorphism F_(U_i∩ U_j)_0^*Ω^1_(U_i∩ U_j)_0→_(U_i∩ U_j)_0. The cocycle {s_ij} is a representative of κ in  H^1(X_0,(F_X_0^*Ω_X_0,_X_0))=H^1(X_0,F_X_0^*TX_0).

The Deligne–Illusie class <cit.> (see also <cit.>), which is sometimes treated in the literature as analogous to the Kodaira–Spencer class, also obstructs the lift of Frobenius, and it is indeed equivalent to the above class. We can construct the Deligne–Illusie class as a Čech class.  Cover X with affine opens U_i which carry lifts of semi-linear Frobenius F_U_i U_i→ U_i. One defines a 1-cocycle on X by 

    c_ij=F^*_U_i-F^*_U_j_U_i∩ U_j→ pF_*_U_i∩ U_j

where F is some lift of Frobenius on U_i∩ U_j (the subsheaf pF_*_U_i∩ U_j is independent of choices).
Then c_ij(U_i∩ U_j)→(U_i∩ U_j) factors through the differential d in the sense that there is a cocycle 

    h_ijΩ^1_X_0(U_i∩ U_j)→ (F_X_0)_*_X_0(U_i∩ U_j)

that fits into a commutative diagram

    _X[d]_π_0[r]^c_ij    p F_*_X
    _X_0[d]_d   
    Ω^1_X_0[r]^>>>>>h_ij[r]   (F_X_0)_*_X_0[uu]_p

This cocycle h_ij gives an element of 

    H^1(X_0,(Ω^1_X_0,(F_X_0)_*_X_0))=H^1(X_0,(F_X_0^*Ω_X_0^1,_X_0)).



 The Deligne–Illusie class is also defined using lifts of relative Frobenius F and written as a element of H^1(X^(p)_0,(Ω^1_X^(p)_0,F_*_X_0)) where X_0^(p) is the base-change by Frobenius, X_0×_. To see that  this is equivalent, we relate the  sheaves of differentials that appear. The absolute Frobenius on X_0 factors as

    X_0[r]^F   X_0^(p)[r]^G   X_0

where G is the isomorphism induced by the base-change by Frobenius on .
From the exact sequence 

    0[r]   G^*Ω^1_X_0[r]   Ω^1_X_0^(p)[r]   Ω^1_X_0^(p)/X_0[r]   0

and the vanishing of Ω^1_X_0^(p)/X_0, we have an isomorphism between G^*Ω^1_X_0 and Ω^1_X_0^(p). Pulling each back by F, we get

    F_X_0^*Ω^1_X_0≅ F^*Ω^1_X_0^(p).




We have the equality up to sign of the extension class and the Deligne–Illusie class κ=-h.



The Frobenius lift F_U_i^*(U_i)→(U_i) is equivalent to a splitting h_iΩ^1,_U_i→_(U_i)_0 over U_i by

    F^*_U_i(x)=x^p+h_i(d^x).

This splitting induces σ_i F_U_i^*Ω^1_(U_i)_0→Ω^1,_U_i 
by σ_i(F_U_i^*dx_0)=d^ x-α(h_i(d^ x)) for any lift x of x_0.
Consequently, for x∈(U_i∩ U_j), we have 

    α(F^*_U_i(x)-F^*_U_j(x))=α(h_i(d^x)-h_j(d^x))=σ_j(F_U_j^*dx_0)-σ_i(F_U_i^*dx_0)

giving the equality of the Deligne–Illusie and the extension classes.




§ ARITHMETIC GAUSS–MANIN HOMOMORPHISM


There is an arithmetic Gauss–Manin homomorphism in our framework. It is connecting homomorphism  H^0(X_0,F_X_0^*Ω_X_0^1)→ H^1(X_0,_X_0) of the long exact sequence attached to 

    0[r]    _X_0[r]   Ω_X^1,[r]   F_X_0^*Ω^1_X_0[r]   0.

This is analogous to the Gauss–Manin connection for the first de Rham cohomology over a one-dimensional base g Y→ S arising from the exact sequence

    0[r]    g^*Ω^1_S[r]   Ω_Y^1[r]   Ω^1_Y/S[r]   0.




The arithmetic Gauss–Manin homomorphism is given by cup product with the extension class κ.



We compute the arithmetic Gauss–Manin homomorphism. Let ω∈ H^0(X_0,F_X_0^*Ω_X_0^1). We pick a covering U_i of X such that  on U_i, ω_i lifts to 
ω_i∈Ω^1,_U_i. Then, the image of the Gauss–Manin homomorphism is the cocycle g_ij=ω_i-ω_j. Because ω_i=ω_j on U_i∩ U_j, g_ij is valued in _X_0.

By refining the cover, we may suppose that we have a splitting of the fundamental exact sequence on each U_i, σ_i F_(U_i)_0^*Ω_(U_i)_0^1→Ω_U_i^1,. Set ω_i=σ_i(ω_i)
Then the image of the Gauss–Manin map is given by the cocycle g_ij=σ_i(ω_i)-σ_j(ω_j). On U_i∩ U_j we have

    g_ij   =   σ_i(ω_i)-σ_j(ω_j)
       =   σ_i(ω_i)-σ_j(ω_i)
       =   (σ_i-σ_j)(ω_i)
       =   s_ij(ω_i).

Therefore, by unwinding the definition of the cup product, we see 

    {g_ij}={ω_i}∪{s_ij}.


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