metadata
tags:
- sentence-transformers
- sentence-similarity
- feature-extraction
- generated_from_trainer
- dataset_size:264888
- loss:CosineSimilarityLoss
base_model: sentence-transformers/all-MiniLM-L6-v2
widget:
- source_sentence: >-
latex_in_original_or_summarized: K(M, n)
[SEP]
summarized: $K(M, n)$
[SEP]
main_note_content: Chain complexes and spaces. [59], that for
simplicial sheaf $\text{X}$ we denote by $C_{*}(\mathcal{X})$ the
(normalized) chain complex $C_{*}(\mathcal{A}$ associated to the
sheaf abelian groups $\mathbb{X}$. This defines a functor
$$ C_{*}: \Delta^{o p} S h v_{N i s}\left(S m_{k}\right) C_{*}(\text{A}
b(k)) $$$ ^f7eebc
which is well (see $[44,59]$ instance) to have a right adjoint
6.2 \mathbb{A}^{1}$-Derived Category Spaces
161
$$ K: C_{*}(\mathcal{A} b(k)) \rightarrow \phi^{o p} S h v_{N i
s}\left(S $$
called the space
For an abelian $M b(k)$ and an integer $n$ we define the pointed
simplicial sheaf $K(M, n)$ (see [59, page 56]) $K$ to the shifted
complex $M[n]$, the complex $M$ placed in degree 0 . If n< 0, the space
$K(M, n)$ is a point. If $n \geq 0$ then $K(M, n)$ has only one
non-trivial sheaf which is the and which is canonically isomorphic to
$M$. More generally, for a chain $C_{*}$, $K C_{*}$ has homotopy
sheaf 0 $n< 0$, and the $n$-th homology sheaf $H_{n}\left(C_{*}\right)$
for $n \geq 0$.
It is clear that $C_{*}: \Delta^{o p} S h i s}\left(S m_{k}\right)
\rightarrow b(k))$ sends simplicial weak equivalences to
quasi-isomorphisms and $K: C_{*}(A b(k)) \rightarrow \Delta^{o p} S h v_{N
i s}\left(S m_{k}\right)$ maps quasi-isomorphisms to simplicial
equivalences. If $C_{*}$ fibrant, it follows that $K\left(C_{*}\right)$
is simplicially Thus the two functors induce a pair of adjoint functors
$$ C_{*}: \mathcal{H}{s}(k) \rightarrow D(\mathcal{A} b(k)) $$ ^c4a825
$$ K: D(\mathrm{A} b(k)) \rightarrow \mathcal{H}_{s}(k) $$
As a consequence it is clear that is an $\mathscr{A}^{1}$-local
complex, space $K\left(C_{*}\right)$ is an $\mathbb{A}^{1}$-local space.
Thus $C_{}: \mathbf{H}_{s}(k) \rightarrow maps $\mathcal{A}^{1}$-weak
to $\mathrm{A}^{1}$-quasi and induces a functor
\rightarrow D_{\mathbb{A}^{1}}(A b(k)) $$
which in concrete terms, maps a space $\operatorname{X}$ to the
$\mathbb{A}^{1}$-localization of $C_{*}(\mathcal{X})$. We denote the
latter by $C_{*}^{A^{1}}(\mathbb{X})$ and call it the
$\mathbb{A}^{1}$-chain of $\mathcal{X}$. functor
$C_{*}^{\operatorname{A}^{1}}: \mathfrak{H}(k) \rightarrow b(k))$ admits
as right adjoint the functor $K^{\mathbb{A}^{1}}:
D_{\mathbb{A}^{1}}(\mathcal{A} b(k)) \rightarrow \mathcal{H}(k)$ induced
by $C_{*} \mapsto K\left(L_{\mathbb{A}^{1}}\left(C_{*}\right)\right)$. We
that for an $\mathbb{A}^{1}$-local complex the space
$K\left(C_{*}\right)$ is automatically $\mathbb{A}^{1}$-local and thus
simplicially equivalent to the space
[SEP]
processed_content: the pointed simplicial where $M$ \in b(k)$ and $n$
is integer. It is defined by applying to the complex $M[n]$, of the
complex degree 0 .
sentences:
- >-
latex_in_original_or_summarized: \gamma_1=(m_1,N_1,a_1)
[SEP]
summarized: $\gamma_1=(m_1,N_1,a_1)$
[SEP]
main_note_content: \begin{notation}\label{Dep1}
Let $\gamma_1=(m_1,N_1,a_1)$, $\gamma_2=(m_2,N_2,a_2)$ be an ordered
pair of
(generalized) monodromy data which hypothesis (A). Assume that
$m_1|m_2$.
Set $d:=m_2/m_1$ and $r:=\gcd(m_1, a_1(N_1))$.
Then, \eqref{Dep} to
$\epsilon=d(r-1)$ and $g_3=dg_1+g_2+\epsilon$.
In particular, $\epsilon=0$ if and if $r=1$.
\end{notation}
[SEP]
processed_content:
- >-
latex_in_original_or_summarized: \langle u\rangle G W(F)
[SEP]
summarized: $\langle u\rangle \in G W(F)$
[SEP]
main_note_content: Let us denote (in characteristic) by $G W(F)$ the
Grothendieck-Witt ring of isomorphism classes of non-degenerate
symmetric bilinear forms [48]: this is the group completion of the
commutative monoid of isomorphism classes of non-degenerate symmetric
forms for the direct sum.
For $u \in F^{\times}$, we denote by $\langle u\rangle G W(F)$ the form
on vector space of rank one given by $F^{2} F,(x, \mapsto u x y .$
By the results of loc. \langle u\rangle$ generate $G as a group. The
following Lemma is (essentially) [48, Lemma (1.1) Chap. IV]:
[SEP]
processed_content:
- >-
latex_in_original_or_summarized: $\varepsilon_{\infty}$
[SEP]
summarized: $\varepsilon_{\infty}$
[SEP]
main_note_content: To compute the genus of $X(\kappa)$, further
specialize to $\Gamma_{1}=\Gamma$ and $\Gamma_{2}=$
$\mathfrak{SL}_{2}(\mathbb{Z}) . Let $y_{2}=\mathrm{SL}_{2}(\mathbb{Z})
i, y_{3}=\mathrm{SL}_{2}(\mathbb{Z}) \mu_{3}$, and
$y_{\infty}=\mathfrak{SL}_{2}(\mathbb{Z}) \infty$ be the elliptic point
of period 2, the elliptic point of period 3, and the cusp of $X(1)=$
SL_{2}(\mathbb{Z}) \backslash \mathcal{H}^{*} .$ Let $\varepsilon_{2}$
and $\varepsilon_{3}$ be the number of elliptic points of $\Gamma$ in
$f^{-1}\left(y_{2}\right)$$ and of^{-1}\left(y_{3}\right)$, i.e., the
number of elliptic points of period 2 and 3 in $X(\Gamma)$, and let
$\varepsilon_{\infty}$ be the number of cusps of X(\Gamma) .$ Then
recalling that $d=\operatorname{deg}(f)$ and letting $h=2$ or $h=3$, the
formula for $d$ at the beginning of the section and then the formula for
$e_{\pi_{1}(\tau)}$ at the nonelliptic points and the elliptic points
over $\mathrm{SL}_{2}(\mathscr{Z}) y_{h}$ show that (Exercise 3.1.3(a))
$$ d=\sum_{x \in f^{-1}\left(y_{h}\right)} e_{x}=h
\cdot\left(\left|f^{-1}\left(y_{h}\right)\right|-\varepsilon_{h}\right)+1
\cdot \varepsilon_{h} $$
and using these equalities twice gives
$$ \sum_{x \in
f^{-1}\left(y_{h}\right)}\left(e_{x}-1\right)=(h-1)\left(\left|f^{-1}\left(y_{h}\right)\right|-\varepsilon_{h}\right)=\frac{h-1}{h}\left(d-\varepsilon_{h}\right)
$$
$68 \quad 3$ Dimension Formulas
Also.
$$ \sum_{x \in
f^{-1}\left(y_{\infty}\right)}\left(e_{x}-1\right)=d-\varepsilon_{\infty}
$$
Since $X(1)$ has genus 0, the Riemann-Hurwitz formula now shows
[SEP]
processed_content:
- source_sentence: >-
latex_in_original_or_summarized: $M_\ell(C \to S) = M_\ell(S)$
[SEP]
summarized: $M_\ell(C \to S) = M_\ell(S)$
[SEP]
main_note_content: If $C \to S$ is a relative smooth proper curve of genus
$g \geq 1$ over an irreducible base, then the $\ell$-torsion of relative
Jacobian of $C$ information about the family. Suppose $\ell$ is
invertible on $S$, and let \in S$ be a geometric point. The fundamental
group $\pi_1(S,s)$ acts
linearly on the fiber $\operatorname{Pic}^0(C)[\ell]_{s} \cong
(\mathbb{Z}/\ell)^{2g}$,
one can consider the mod-$\ell$ representation associated to $C$:
$$\rho_{C \to S, \ell}:\pi_1(S,s) \rightarrow \cong
\operatorname{GL}_{2g}(\mathbb{Z}/\ell).$$ ^e59a92
Let $M_\ell(C \to S)$, or simply $M_\ell(S)$, be the image
of this representation.
If a primitive $\ell$th root of is defined $S$, then
$\operatorname{Pic}^0(C)[\ell]_{s}$ is equipped
with a skew-symmetric form $\langle \cdot,\cdot and $M_\ell(C \to S)
\subseteq
\operatorname{Sp}(\operatorname{Pic}^0(C)[\ell]_s,\langle \rangle) \cong
\operatorname{Sp}_{2g}(\mathbb{Z}/\ell)$.
If C \to S$ is a sufficiently general family of curves, then
$M_\ell(C \to S) \cong \operatorname{Sp}_{2g}(\mathbb{Z}/\ell)$
\cite{delignemumford}.
In this we compute when $S$ is an irreducible component of moduli space
of hyperelliptic or trielliptic curves and $C \to S$ is the tautological
curve. The first result implies that there is no restriction on the
monodromy group in the hyperelliptic case other than that it preserve the
symplectic pairing. As trielliptic curve is a $\mathbb{Z}/3$-cover of a
genus zero curve, the $\mathbb{Z}/3$-action constrains the monodromy
group to lie in a unitary group associated to $\mathbb{Z}[\zeta_3]$. The
second result implies that this is the only additional restriction in the
trielliptic case.
\paragraph{Theorem \ref{thhe}}
{\it
$\ell$ be an odd prime, and let $k$ be an closed in which $2\ell$ is invertible.
For $g\geq 1$, $M_\ell(\mathcal{H}_g\otimes k)\cong
\operatorname{Sp}_{2g}(\mathbb{Z}/\ell)$.}
\paragraph{Theorem \ref{thtri}}
{\it
Let $\ell\geq 5$ be prime, and let $k$ be closed field in which $3\ell$
is invertible.
$\mathcal{T}^{\bar\gamma}$ be any component the moduli space
trielliptic curves of genus $g\geq Then
$M_\ell(\mathcal{T}^{\bar\gamma}\otimes k) \cong
\operatorname{SG}_{(r_\gamma,s_\gamma)}(\mathbb{Z}/\ell)$ (where the
latter is unitary group defined
in \eqref{eqdefsg}).}
\medskip
We also prove that the $\ell$-adic monodromy group
$\operatorname{Sp}_{2g}(\mathbb{Z}_\ell)$ in the situation of Theorem
\ref{thhe} and is
$\operatorname{SG}_{(r_\gamma,s_\gamma)}(\mathbb{Z}_\ell)$
in the of Theorem \ref{thtri}.
Theorem \ref{thhe} is an unpublished result J.K. Yu and has already been
used multiple times in literature.
In \cite{chavdarov}, Chavdarov assumes this result show that the
numerator of the zeta function of
the typical hyperelliptic curve over a finite field is irreducible.
Kowalski also uses this result in a similar fashion \cite{kowalskisieve}.
The first author used Theorem to prove a conjecture of and
Washington on class of quadratic function fields
There are other results in the literature which similar to Theorem
\ref{thhe}
but which are not quite strong enough for the above.
A'Campo \cite[Th.\ 1]{acampo} computes the topological of $\mathcal{H}_g
\otimes
On the arithmetic side, the $\mathbb{Q}_\ell$,
as opposed to $\mathbb{Z}_\ell$, monodromy of $\mathcal{H}_g$
is computed in \cite[10.1.16]{katzsarnak}. Combined with a theorem of
Larsen on compatible families of representations \cite[3.17]{larsenmax},
this shows that the mod-$\ell$ group
of $\mathcal{H}_g$ is maximal for a set of
primes $\ell$ of density one (as opposed to for all $\ell \geq 3$).
There are results on $\mathbb{Q}_\ell$-monodromy cyclic covers of the
projective
line of arbitrary degree, e.g., \cite[Sec. 7.9]{katztwisted}. Also,
in \cite[5.5]{fkv}, the authors prove that the projective representation
$\mathbb{P} \rho_{C \to S,\ell}$ surjective for many
families of cyclic covers the projective line.
Due to a combinatorial their theorem does not apply to $\mathcal{H}_g$
and applies to at most one component of the moduli space of
trielliptic curves for each see Remark \ref{Rfkv}.
See also work of Zarhin, e.g., \cite{zarhincyclic}.
an application, for all $p \geq show using
exist hyperelliptic and trielliptic curves
of every genus signature) defined over $\bar{\mathbb{F}}_p$ whose
Jacobians absolutely simple.
In contrast with the applications above,
these corollaries do not use the full strength of our results.
Related can be found in \cite{HZhu} authors produce curves with
absolutely
Jacobians over $\mathbb{F}_p$ under the $g \leq 3$.
\paragraph{Corollary \ref{Chypabsirr}}
{\it Let p \not = 2$ let Then there exists a
smooth hyperelliptic curve of genus $g$ over $\bar{\mathbb{F}}_p$ whose
Jacobian is
absolutely simple.}
\paragraph{Corollary \ref{Ctriabsirr}}
{\it Let $p \not = 3$. $g 3$ and be a trielliptic signature for $g$
\ref{Dtrisig}).
Then there exists a smooth trielliptic curve defined over with genus $g$
and signature $(r,s)$
whose Jacobian is simple.}
\medskip
Our proofs proceed by induction on the genus.
The base cases for the family
rely on the fact that every curve of genus $g=1,2$ is hyperelliptic;
the claim on monodromy follows from the analogous assertion the monodromy
of $\mathcal{M}_g$.
The case for the trielliptic family involves a comparison with
a Shimura variety of PEL type, namely, the modular variety.
An important step is to show the monodromy group does not change in the
base cases when
one adds a labeling of the ramification points to the moduli problem.
The step is similar to the method used in \cite{ekedahlmono}
and uses the fact that families of smooth hyperelliptic (trielliptic)
curves degenerate to trees of (trielliptic) curves of lower genus.
The combinatorics of admissible degenerations require us
to compute the monodromy exactly for the inductive step rather than up to
isomorphism.
The inductive strategy using admissible degeneration developed here
should work for other of curves, especially for more general
cyclic covers of projective The difficulty is in direct
calculation of monodromy for the necessary base cases.
We thank C.-L.\ Chai, R.\ Hain, A.J.\ de Jong, E. Kani, and J. Kass.
[SEP]
processed_content: the image of the mod-$\ell$ representation $\rho_{C
\to \ell}$ of the relative smooth $C \to S$ of genus $g \geq 1$ over an
irreducible base.
sentences:
- >-
latex_in_original_or_summarized: X^{\vee}
[SEP]
summarized:
[SEP]
main_note_content: Let be principally polarized abelian scheme of
relative dimension $g$ over an irreducible base.
If $\ell$ is a
rational invertible on $S$, then the $\ell$-torsion $X[\ell]$ of
$\ell$ is an \'etale cover of with geometric fiber isomorphic to
$(\mathbb{Z}/\ell)^{2g}$.
Let $s$ be a geometric point of $S$. The group $\pi_1(S,s)$
linearly on the $\ell$-torsion of $X$.
This yields a representation
\rho_{X \to S, s,\ell}: \pi_1(S,s) \rightarrow
\operatorname{Aut}(X[\ell]_s) \cong
\operatorname{GL}_{2g}(\mathbb{Z}/\ell).$$ ^dbec50
The cover $X[\ell] \to S$ both determines and is determined by
representation \to S, s,\ell}$.
The image of \to S, is the mod-$\ell$ monodromy of $X \to S$ and we
denote it by $M_\ell(X \to S, s), or by $M_\ell(S,s)$ if the choice of
abelian scheme is clear.
The isomorphism class of the
$M_\ell(S,s)$ is independent of the choice of base point $s$,$ and we
denote it $M_\ell(S)$.
Let $X^{\vee}$ be the dual abelian scheme. There a pairing $X[\ell]
\times X^{\vee}[\ell] \to \boldsymbol{\mu}_{\ell,S}$, where :=
\boldsymbol{\mu}_\ell \times S$ is group scheme of $\ell\th$ of unity.
polarization induces an isomorphism $X \to X^{\vee}$, and
thus a skew-symmetric pairing $X[\ell] \times X[\ell] \to
\boldsymbol{\mu}_{\ell,S}$.
Because the polarization is defined globally, the image of monodromy
$M_\ell(X \to S, s)$ is contained in the group of symplectic
similitudes of $(X[\ell]_s,
\langle \rangle_\phi)$, which is isomorphic to
$\operatorname{GSp}_{2g}(\mathbb{Z}/\ell)$. Moreover, if a primitive
$\ell^{{\rm root of
unity globally on $S$, $\pi_1(S,s)$ acts trivially on
$\boldsymbol{\mu}_{\ell,S}$ and $M_\ell(X \to S,s) \subseteq
\cdot,\cdot \rangle_\phi) \cong \operatorname{Sp}_{2g}(\mathbb{Z}/\ell).
Similarly, the $X[\ell^n] S$ defines a monodromy representation
with in $\operatorname{Aut}(X[\ell^n]_s)
\cong\operatorname{GL}_{2g}(\mathbb{Z}/\ell^n)$. Taking
inverse limit over all n, we obtain a continuous representation on the Tate module of $X$,
$$\rho_{X \to S, s}: \pi_1(S,s) \rightarrow \varprojlim_n
\operatorname{Aut}(X[\ell^n]_s) \cong
\operatorname{GL}_{2g}(\mathbb{Z}_\ell).$$
^f6240a
We denote the image of this representation by $M_{\mathbb{Z}_\ell}(X
\to and its isomorphism class by $M_{\mathbb{Z}_\ell}(X \to S)$ or
$M_{\mathbb{Z}_\ell}(S)$.
Again, there is an
M_{\mathbb{Z}_\ell}(X \to S) \subseteq
If
$F$ is a field, let $F_{\ell^\infty} =
F(\boldsymbol{\mu}_{\ell^\infty}(\bar F))$. If $S$ is an then
$$M_{\mathbb{Z}_\ell}(X \to S, s)/ F} \to S \otimes{\bar F}, s) \cong
^dd1bab
Finally, let $M_{\mathbb{Q}_\ell}(X\to$ S, s)$ be the Zariski closure
of \to S, s)$ in $\operatorname{GL}_{2g}(\mathbb{Q}_\ell)$.
Now suppose that \psi:C \to S$ is a relative proper semi-stable curve.
Let $\operatorname{Pic}^0(C) := \operatorname{Pic}^0_{C/S}$ be the
neutral component of the relative Picard of $C$ over $S$. Since $C/S$
semi-stable, $\operatorname{Pic}^0(C)$ is a semiabelian scheme
[[bosch_lutkebohmert_raynaud_nm_Theorem
1_page_259|\cite[9.4.1]{blr}]].
Suppose that there is least one geometric point such the fiber
$\operatorname{Pic}^0(C_s)$ is an abelian variety. (This is true[^5] if
some $C_s$ is a tree smooth curves.) Then there is a nonempty open
subscheme $S^*$ of $S$ such that $\operatorname{Pic}^0(C|_{S^*})$ an
abelian scheme over $S^*$.
[^5]: cf. Abelian varieties isogenous to a Jacobian by CL Chai, which
talks about a tree of smooth curves having a Jacobian that is an abelian
variety that is actually the product of the Jacobians of irreducible
We define the mod-$\ell$ and $\mathbb{Z}_\ell$ monodromy representations
of $C$ to be those of $\operatorname{Pic}^0(C|_{S^*}) \to S^*$.
(Alternatively, may constructed as the restrictions of
$R^1\psi_*\boldsymbol{\mu}_{\ell,S}$ and
$R^1\psi_*\boldsymbol{\mu}_{\ell^\infty,S}$ largest subscheme of $S$
on which these sheaves are unramified.)
Thus, $M_\ell(C \to s) = M_\ell(\operatorname{Pic}^0(C|_{S^*}) \to S^*,
s)$, and we denote this again by M_\ell(S,s) if the curve is clear and
by the base point is suppressed. ^37a851
The moduli spaces $\overline{\mathcal{M}}_G$ and
$\widetilde{\mathcal{M}}_G$ are Deligne-Mumford stacks, and we employ a
similar formalism for \'etale covers of stacks \cite{noohi}.
$\mathcal{S}$ a connected Deligne-Mumford The category of Galois \'etale covers of $\mathcal{S}$ is a Galois category the sense of Grothendieck, and thus there is \'etale fundamental
of More precisely, let $s\in \mathcal{S}$ be a geometric
Then there is a group $\pi_1(\mathcal{S},s)$ and an equivalence of
between finite $\pi_1(\mathcal{S},s)$-sets$ and finite \'etale Galois
covers of $\mathcal{S}$.
If $\mathcal{S}$ has a coarse moduli space $S_{\mathrm{mod}}$, then
$\pi_1(\mathcal{S},s)$ is the extension of $\pi_1(S_{\mathrm{mod}},s)$
by a group which encodes extra automorphism structure on the moduli
space S_{\mathrm{mod}} [[noohi_fgas_thm 7.11|\cite[7.11]{noohi}]].
If $X \to \mathcal{S}$ is a family of abelian varieties, we again let
$M_\ell(X\to be the of $\pi_1(\mathcal{S}, s)$ in ^758472
Let $\mathcal{C}^\gamma$ be the tautological labeled curve over
By the mod-$\ell$ or $\mathbb{Z}_\ell$ monodromy of
$\widetilde{\mathcal{M}}_G^\gamma$ we mean of $C^\gamma \to
\widetilde{\mathcal{M}}_G^\gamma$. [^6]
[^6]: #_meta/TODO/question that that $C^\gamma \to
\widetilde{\mathcal{M}}_G^\gamma$ gets to have relative Picard group of
its own? How does that make sense when
$\widetilde{\mathcal{M}}_G^\gamma$ a is not a scheme?
[SEP]
processed_content: the dual abelian scheme of the abelian scheme $X/S$.
There is a canonical pairing $X[\ell] \times X^{\vee}[\ell] \to
\boldsymbol{\mu}_{\ell,S}$, where $\boldsymbol{\mu}_{\ell,S} :=
\boldsymbol{\mu}_\ell \times S$ is group scheme of $\ell\th$ roots of
unity.
- >-
latex_in_original_or_summarized: \mathbb{Th}_f \phi
[SEP]
summarized: $_f
[SEP]
main_note_content: It be convenient to work in stable category
$\mathcal{Spt}(B)$$ of $P^1$-spectra over $B$, where $B$ is a finite
type scheme over frequently, $B=L$, where $L$ is a field extension of
$k$.
The notation be the morphisms. $(B)$ is a monoidal category under
smash product $\wedge$, with $1_B$, denoting the sphere spectrum.
Any pointed simplicial presheaf $X$ determines corresponding
$\mathbb{P}^1$-suspension spectrum $\Sigma^{\infty} X$.
For $\Sigma^{} Spec L_+ 1_L$ and $\Sigma^{\infty} (^1_L)^{ n}$ is a
suspension When working in $\operatorname{Spt}(L)$, we will identify
pointed $X$ with their spectra $\Sigma^{} X$, omitting the
$\Sigma^{\infty}$. ^1246cf
We will use six operations $(p^*, p_!, p^!, \wedge, given by Ayoub
developed by Ayoub, and Cisinksi-Déglise
\cite{CD-triang_cat_mixed_motives}. There a nice summary in \cite[\S
We use following associated notation and constructions.
When \to Y$ is smooth, $p^*$ admits a left denoted p_{\sharp},
induced by forgetful functor \to \operatorname{Sm}_{Y}$ from smooth
over $X$ smooth schemes over $Y$.
For $p:X\to \operatorname{Spec} L$ a smooth scheme over $L$, the
suspension spectrum of $X$ is canonically identified with as an object
of $\operatorname{Spt}(L)$.
For a vector bundle $p:E \to X$, the Thom spectrum Th(E)$ (or just is
canonically identified $s^*p^! 1_X$[^2].
Perhaps $s$ a fixed section of $p$.$
Let $\Sigma^E$ equal $\Sigma^E = s^* p^!: (X) \to (X)$. Let $e: \to X
and $d: D Y$ be two vector bundles over smooth $p: X L$ and $q:Y
\operatorname{Spec} L$. ^123eb1
Given a map $f: Y \to X$ and a monomorphism $\phi: D \hookrightarrow
f^* there is an associated natural transformation ^0f1ba8
$$_f \phi : q^! p_! \Sigma^E p^!$$
of endofunctors on $(L)$ inducing the map on Thom spectra. The \phi$
is defined as composition ^0b33ea
\begin{equation}\operatorname{Th}_f = {1_{f^*E}} \circ
.\end{equation}$$
The natural $\operatorname{Th}_{1_Y} is the composition t^*d^! t^*
^!e^!\to t^* \phi^* e^! \cong e^!,$$ where $t: D$ denotes the zero
section of $D$, $s: X \to E$ denotes the zero $E$, and the middle
arrow is by the exchange transformation $\phi^! \cong \to 1^! \phi^*
\cong natural transformation $\operatorname{Th}_f the composition
$$\begin{equation}\operatorname{Th}_f 1: q_! \Sigma^{f^* E} q^! \cong
p_! f^! p^! \cong p_!^E f_! f^! p^! {\rightarrow} p_! ^E
p^!,\end{equation}$$
where $: f_! f^! \to 1$ denotes the counit.
[SEP]
processed_content:
- "latex_in_original_or_summarized: j_0: \\mathbb{G}_m / \\bar{k} \\subset \\mathbb{A}^1 / \n\n[SEP]\n\nsummarized: $j_0$\n\n[SEP]\n\nmain_note_content: In order to explain the simple underlying ideas, we will admit four statements, and explain how to deduce from them equidistribution theorems about the sums $S(M, k, \\chi)$ as $\\chi$ varies.\n\n(1) If $M$ and $N$ are both perverse on $\\mathbb{G}_m / k$ (resp. on $\\mathbb{G}_m / \\bar{k}$ ) and satisfy $\\mathcal{P}$, then their middle convolution $M _{\\text {mid }} N$ is perverse on $\\mathbb{G}_m / k$ (resp. on $\\mathbb{G}_m / \\bar{k}$ ) and satisfies $\\mathcal{P}$.\n\n(2) With the operation of middle convolution as the \"tensor product,\" the skyscraper sheaf $\\delta_1$ as the \"identity object,\" and $[x \\mapsto 1 / x]^{\\star} D M$ as the \"dual\" $M^{\\vee}$ of $M$ ( $D M$ denoting the Verdier dual of $M$ ), the category of perverse sheaves on $\\mathbb{G}_m / k$ (resp. on $\\mathbb{G}_m / \\bar{k}$ ) satisfying $\\mathcal{P}$ is a neutral Tannakian category, in which the \"dimension\" of an object $M$ is its Euler characteristic $_c\\left(_m / , M\\right)$.\n\n(3) Denoting by\n\n$$ j_0: \\mathbb{G}_m / \\bar{k} \\subset \\mathbb{A}^1 / \\bar{k} $$ ^212b11\n\n1. OVERVIEW\n\n11\n\nthe inclusion, the construction\n\n$$ M \\mapsto H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!} M\\right) $$ ^425e70\n\nis a fibre functor on the Tannakian category of perverse sheaves on $\\mathbb{G}_m / \\bar{k}$ satisfying $\\mathcal{P}$ (and hence also a fibre functor on the subcategory of perverse sheaves on $\\mathbb{G}_m / k$ satisfying $\\mathcal{P}$ ). For $i \\neq 0, H^i\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!} M\\right)$ vanishes.\n\n(4) For any finite extension field $E / k$, and any multiplicative character $\\rho$ of $E^{\\times}$, the construction\n\n$$ M \\mapsto H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(M \\otimes \\mathcal{L}_\\rho\\right)\\right) $$ ^f07855\n\nis also such a fibre functor. For $i \\neq 0, H^i\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(M \\otimes \\mathcal{L}_\\rho\\right))$ vanishes.\n\nNow we make use of these four statements. Take for $N$ a perverse sheaf on $\\mathbb{G}_m / k$ which is $\\iota$-pure of weight zero and which satisfies $\\mathcal{P}$. Denote by $\\langle N\\rangle_{ {arith }}$ the full subcategory of all perverse sheaves on $\\mathbb{G}_m / k$ consisting of all subquotients of all \"tensor products\" of copies of $N$ and its dual $N^{\\vee}$. Similarly, denote by $\\langle N\\rangle_{ {geom }}$ the full subcategory of all perverse sheaves on $\\mathbb{G}_m / \\bar{k}$ consisting of all subquotients, in this larger category, of all \"tensor products\" of copies of $N$ and its dual $N^{\\vee}$. With respect to a choice $\\omega$ of fibre functor, the category $\\langle N\\rangle_{\\text {arith }}$ becomes[^5] the category of finite-dimensional $\\overline{\\mathbb{Q}}_{\\ell}$-representations of an algebraic group $G_{a r i t h, N, \\omega} \\subset G L(\\omega(N))=G L('\\operatorname{dim}' N)$, with $N$ itself corresponding to the given \" dim\" $N$-dimensional representation. Concretely, $G_{arith,N, \\omega} \\subset G L(\\omega(N))$ is the subgroup consisting of those automorphisms $\\gamma$ of $\\omega(N)$ with the property that $\\gamma$, acting on $\\omega(M)$, for $M$ any tensor construction on $\\omega(N)$ and its dual, maps to itself every vector space subquotient of the form $$ (any subquotient of $$ ).\n\n[^5]: Recall that associated to a neutral Tannakian category $(C, \\omega)$ is an affine algebraic group $G$ (called the Tannakian group or Tannakian dual of the neutral Tannakian category) and the fiber functor $\\omega$ induces an equivalence $C \\to \\operatorname{Rep}(G)$ of tensor categories, so $G_{\\text{arith}, N, \\omega}$ is being defined as this algebraic group for $\\langle N \\rangle_{\\text{arith}}$ under the choice of $\\omega$.\n\n^370dc9\n\nAnd the category $\\langle N_{\\text {geom }}$ becomes the category of finite-dimensional $\\overline{\\mathbf{Q}}_\\ell$-representations of a possibly smaller algebraic group $G_{\\text{geom}, N, \\omega} \\subset G_{\\text {arith }, N, \\omega}$ (smaller because there are more subobjects to be respected).\n\nFor $\\rho$ a multiplicative character of a finite extension field $E / k$, we have the fibre functor $\\omega_\\rho$ defined by\n\n$$ M \\mapsto H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{!}\\left(M \\mathcal{L}_\\rho\\right)\\right) $$\n\non $\\langle N\\rangle_{\\text {arith }}$. The Frobenius $\\operatorname{Frob}_E$ is an automorphism of this fibre functor, so defines an element $\\operatorname{Frob}_{E, \\rho}$ in the group $G_{a r i t h, N, _\\rho}$ defined[^5] by this choice of fibre functor. But one knows that the groups $G_{\\text {arith }, N, \\omega}$ (respectively the groups $G_{g e o m, N, \\omega}$ ) defined by different fibre functors are pairwise isomorphic, by a system of isomorphisms which are unique up to inner automorphism of source (or target). Fix one choice, say\n\n12\n\n1. OVERVIEW\n\n$\\omega_0$, of fibre functor, and define\n\n$$ G_{\\text {arith }, N}:=G_{\\text {arith }, N, \\omega_0}, \\quad G_{\\text {geom }, N}:=G_{\\text {geom }, N, \\omega_0} . $$\n\nThen the element $Frob_{E, \\rho}$ in the group $G_{\\text {arith }, N, \\omega_\\rho}$ still makes sense as a conjugacy class in the group $G_{\\text {arith }, N}$.\n\nLet us say that a multiplicative character $\\rho$ of some finite extension field $E / k$ is good for $N$ if, for\n\n$$ j: \\mathbb{G}_m / \\bar{k} \\subset \\mathbb{P}^1 / \\bar{k} $$\n\nthe inclusion, the canonical \"forget supports\" map\n\n$$ R j_1\\left(N \\otimes L_\\right) R j_{\\star}\\left(N \\otimes _\\rho\\right) $$\n\nis an isomorphism. If $\\rho$ is good for $N$, then the natural \"forget supports\" maps\n\n$$ H_c^0\\left(\\mathbb{G}_m / , N \\otimes \\mathcal{L}_\\rho\\right)=H_c^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!}(N \\otimes \\mathcal{L}_\\rho)\\right) \\rightarrow H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(N \\otimes L_\\rho\\right)\\right), $$\n\ntogether with the restriction map\n\n$$ H^0\\left(^1 / \\bar{k}, j_{0!}(N \\otimes \\mathcal{L}_\\rho\\right)) H^0\\left(\\mathbb{G}_m , N _\\rho\\right), $$\n\nare all isomorphisms. Moreover, as $N$ is $$-pure of weight zero, each of these groups is $t$-pure of weight zero.\n\nConversely, if the group $\\omega_\\rho(N):=H^0(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(N \\mathcal{L}_\\rho\\right))$ is $\\iota$-pure of weight zero, then $\\rho$ is good for $N$, and we have a \"forget supports\" isomorphism\n\n$$ H_c^0\\left(\\mathbb{G}_m / \\bar{k}, N \\otimes \\mathcal{L}_\\rho\\right) _\\rho(N):=H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!}\\left(N \\otimes \\mathcal{L}_\\rho\\right)) . $$\n\nThis criterion, that $\\rho$ is good for $N$ if and only if $\\omega_\\rho(N)$ is $\\iota$-pure of weight zero, shows that if $\\rho$ is good for $N$, then $\\rho$ is good for every object $M$ in the Tannakian category $\\langle N\\rangle_{\\text {arith }}$ generated by $N$, and hence that for any such $M$, we have an isomorphism\n\n$$ H_c^0\\left(\\mathbb{G}_m / \\bar{k}, M \\otimes \\mathcal{L}_\\rho\\right) \\cong \\omega_\\rho(M) \\text {. } $$\n\nRecall that geometrically, i.e., on $\\mathbb{G}_m / \\bar{k}$, we may view the various Kummer sheaves $\\mathcal{L}_\\rho$ coming from multiplicative characters $\\rho$ of finite subfields $E \\subset \\bar{k}$ as being the characters of finite order of the tame inertia group $I(0)^{\\text {tame }}$ at 0 , or of the tame inertia group $I()^{ {tame }}$ at $\\infty$, or of the tame fundamental group $_1^{\\text {tame }}\\left(\\mathbb{G}_m / \\bar{k}\\right)$. In this identification, given a character $\\rho$ of a finite extension $E / k$ and a further finite extension $L / E$, the pair $(E, \\rho)$ and the pair ( $L, \\circ N o r m_{L / E}$ ) give rise to the same Kummer sheaf on $\\mathbb{G}_m / \\bar{k}$. Up to this identification of $(E, \\rho)$ with $\\left(L, \\rho \\circ N o r m_{L / E}\\right)$, there are, for a given $N$, at most finitely many $\\rho$ which fail to be good for $N$ (simply because there are at most finitely many tame characters which occur in the local monodromies of $N$ at\n\n1. OVERVIEW\n\n13\n\neither 0 or $$, and we need only avoid their inverses). Indeed, if we denote by $r k(N)$ the generic rank of $N$, there are at most $2 r k(N)$ bad $\\rho$ for $N$.\n\nRecall [BBD, 5.3.8] that a perverse $N$ which is $\\iota$-pure of weight zero is geometrically semisimple. View $N$ as a faithful representation of $G_{\\text {geom,N }}$. Then $G_{\\text {geom,N }}$ has a faithful, completely reducible representation[^7], hence[^6] $G_{\\text {geom,N }}$ is a reductive group. ^260249\n\n[^7]: Apparently, \"completely reducible\" is a synonym for \"semisimple\", cf. https://math.stackexchange.com/questions/334178/definition-completely-reducible-group-representation\n\n[^6]: Milne's algebraic groups, Theorem 22.42 shows that the following are equivalent given a connected algebraic group $G$ over a field of characteristic $0$:\n\t1. $G$ is reductive\n\t2. every finite-dimensional representation of $G$ is semisimple\n\t3. some faithful finite dimensional representation of $G$ is semisimple.\n\tSee also the proof of forey_fresan_kowalski_aftff_3.18 Corollary, which uses this theorem.\n\nLet us now suppose further that $N$ is, in addition, arithmetically semisimple (e.g., arithmetically irreducible). Then $G_{a r i t h, N}$ is also a reductive group. Choose a maximal compact subgroup $K$ of the reductive Lie group $G_{\\text {arith }, N}(\\mathbb{C})$ (where we use $\\iota$ to view $G_{\\text {arith }, N}$ as an algebraic group over $\\mathbb{C}$ ). For each finite extension field $E / k$ and each character $\\rho$ of $E^{\\times}$ which is good for $N$, we obtain a Frobenius conjugacy class $_{E, \\rho}$ in $K$ as follows. Because $\\rho$ is good for $N$, $\\operatorname{Frob}_E$ has, via $\\iota$, unitary eigenvalues acting on $\\omega_\\rho(N)$, i.e., the conjugacy class $\\operatorname{Frob}_{E, \\rho}$ in $G_{\\text {arith }, N}$ has unitary eigenvalues when viewed in the ambient $G L\\left(\\omega_0(N)\\right)$. Therefore its semisimplification in the sense of the Jordan decomposition, $\\operatorname{Frob}_{E, \\rho}^{s s}$, is a semisimple class in $G_{\\text {arith }, N}()$ with unitary eigenvalues. Therefore any element in the class $\\operatorname{Frob}_{E, \\rho}^{s s}$ lies in a compact subgroup of $G_{arith , N}(\\mathbb{C})$ (e.g., in the closure of the subgroup it generates), and hence lies in a maximal compact subgroup of $G_{\\text {arith,N }}()$. All such are $G_{\\text {arith }, N}(\\mathbb{C})$-conjugate, so we conclude that every element in the class $F r o b_{E, \\rho}^{s s}$ is conjugate to an element of $K$. We claim that this element is in turn well-defined in $K$ up to $K$-conjugacy, so gives us a $K$-conjugacy class $\\theta_{E, \\rho}$. To show that $\\theta_{E, \\rho}$ is well-defined up to $K$-conjugacy, it suffices, by Peter-Weyl, to specify its trace in every finite-dimensional, continuous, unitary representation $\\Lambda_K$ of $K$. By Weyl's unitarian trick, every $\\Lambda_K$ of $K$ is the restriction to $K$ of a unique finite-dimensional representation $\\Lambda$ of the $\\mathbb{C}$-group $G_{\\text {arith }, N} / \\mathbb{C}$. Thus for every $\\Lambda_K$, we have the identity\n\n$\\operatorname{Trace}\\left(\\Lambda_K\\left(\\theta_{E, \\rho}\\right)\\right)=\\left(\\Lambda\\left(\\operatorname{Frob} _{E, }^{s s})\\right)=\\operatorname{Trace}\\left(\\Lambda\\left(\\operatorname{Frob} \\theta_{E, \\rho}\\right)\\right)$. ^d42132\n\nWith these preliminaries out of the way, we can state the main theorem.\n\n\n[SEP]\n\nprocessed_content: the inclusion \n\n$$ j_0: \\mathbb{G}_m / \\bar{k} \\mathbb{A}^1 / \\bar{k} $$\n\nThe construction\n\n$$ M \\mapsto H^0\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!} M\\right) $$\n\nis a fibre functor on the Tannakian category of perverse sheaves on $\\mathbb{G}_m / $ satisfying $P$ (and hence also a fibre functor on the subcategory of perverse sheaves on $\\mathbb{G}_m / k$ satisfying $$ ). For $i \\neq 0, H^i\\left(\\mathbb{A}^1 / \\bar{k}, j_{0!} M\\right)$ vanishes."
- source_sentence: >-
latex_in_original_or_summarized: F^i
[SEP]
summarized: $F^i$
[SEP]
main_note_content: no 3 - Examples of and eyact functors -
Let $A$ be a category, $B$ an abelian An additive functor $F: A \rightarrow B called a cohomological functor
CD.
- 21
if for any distinguished ( $\mathrm{X}, \mathrm{Y}, , \mathrm{v}, w$ )
the sequence
$$$ \xrightarrow{F(u)} F(Y) \xrightarrow{F(v)} F(Z) $$
is exact.
The functor $F_0 T^i$ will often be denoted $F^i$. By virtue $l^{}$ axiom
(TR2) triangulated categories, we have the unlimited exact sequence:
$$ \rightarrow F^i(X) \rightarrow F^i(Y) \rightarrow F^ i(Z)
\rightarrow \rightarrow $$ ^a701ca
[SEP]
processed_content: the functor T^i$ $F: A B$ is a cohomological functor
from a triangulated caOtegory to an category. We have the exact sequence
$$ \cdots F^i(X) F^ i(Z) F^{i+1}(X) \rightarrow \cdots $$
sentences:
- >-
latex_in_original_or_summarized: P^*\left(X^*, Y^*\right)=
[SEP]
summarized: $P^*\left(X^*,$ Y^*)
[SEP]
main_note_content: 3.3. Example of exact Let A, A', A" be three
additive categories,
$$ P: A \times A^{\prime} A^{\prime \prime} $$
a bilinear functor additive with respect to each of the arguments
274
- 12 -
C.D.
We then deduce the bilinear
$$ P^*: \times C\left(A^{}) \rightarrow C\left(A^{\prime
\prime}\right) $$
as follows:
Let X^ be an object of $C(A)$ and $Y^\bullet$ be an object of
$P\left(X^\bullet, Y^\bullet\righ.)$ is doublge complex $A^{ }$. We
then set: $P^*(X^\bullet, Y^\bullet\right)=$ simple complex associated
with $\mathbf{P}\left(\mathcal{X}^*,
Let $f$ be a morphism of (resp. $C(A^{}\right)$ ) homotopic to zero and
$Z^*$ be an object (resp. $C(A)$ ). The morphism $P^*(f, Z^*\right)$
(resp. f\right)$ ) is then homotopic zero. We that uniquely defines
a functor:
$$ P^*: K(A) \times K(A^{}\right) K(A^{ \prime}\right) $$
is exact bifunctor.
In particular, let $A$ be additive category. take the functor:
$$ & A^{\circ} \times A A \\ & (X, Y) \leadsto { Hom }(X,
Y) $$
We then obtain by the previous construction a functor
$\mathscr{Hom}^{\circ}: \text{K}()^{} \mathrm{K}(A) \longrightarrow
\mathrm{K}(\mathrm{Ab})$
which, composed with $l_{\mathbb{e functor }} \mathrm{K}(\mathbb{Ab})
\rightarrow \mathrm{Ab}, gives back the fonotor $\mathscr{Hom}_{K(A)}$.
275
[SEP]
processed_content:
- >-
latex_in_original_or_summarized: \pi_1(U)=\pi_1(U,x)
[SEP]
summarized: $\pi_1(U)=\pi_1(U,x)$
[SEP]
main_note_content: We fix a dense affine open $U\subset C$[^2] and an
algebraic closure $k\to\overline{k}$. We fix a geometric point $x\in
U$, that is, an embedding $\mathrm{Spec}(L)\to U$ for $L/k$ an
algebraically-closed extension. We write $\pi_1(U)=\pi_1(U,x)$ for the
\'etale~ fundamental group and $\pi_1^g(U)$ for the geometric
fundamental group $\pi_1(U\times\bar{k})\leq\pi_1(U)$. We fix a set
$\Lambda$ of almost all odd primes $\ell$ which are invertible in $k$.
For each $\ell\in\Lambda$, we fix a lisse flat $\mathbb{Z}_\ell$-sheaf
$\mathcal{L}_\ell\to U$ and let
$\rho_\ell:\pi_1(U)\to\mathrm{GL}_n({\mathbb{Z}_\ell})$ denote the
corresponding representation. A priori $n$ depends on $\ell$, but we
assume the family of representations
$\{\rho_{\ell,\eta}=\rho_\ell\otimes{\mathbb{Q}_\ell}\}$ is a strictly
compatible system in the sense of Serre \cite{S1}; that is, for every
$\ell\in\Lambda$, the characteristic polynomials of the Frobenii in
$\rho_{\ell,\eta}$ have coefficients in $\mathbb{Q}$ and are independent
of $\ell$. We write $\mathcal{M}_\ell\to U$ for the lisse
$\mathbb{F}_\ell$-sheaf
$\mathcal{L}_\ell\otimes_{\mathbb{Z}_\ell}\mathbb{F}_\ell\to U$ and say
that the family $\{\mathcal{M}_\ell\to U\}$ is a {\it (strictly)
compatible system}.
[^2]: ---
detect_regex: []
latex_in_original: ["C/k"]
tags: [_meta/notation_note_named]
---
$C/k$ denotes a proper smooth geometrically connected curve over the
field $k$.
For each $\ell$, we write $G_\ell^a\leq\mathrm{GL}_n(\mathbb{F}_\ell)$
for the image $(\rho_\ell\otimes\mathbb{F}_\ell)(\pi_1(U))$ and
$G_\ell^g\leq G_\ell^a$ for the image of $\pi_1^g(U)$. A priori
$G_\ell^a$ may be any subgroup of $\mathrm{GL}_n(\mathbb{F}_\ell)$, but
if we consider additional arithmetic information, then we may be able to
deduce that $G_\ell^a$ lies in a proper subgroup
$\Gamma_\ell^a\leq\mathrm{GL}_n(\mathbb{F}_\ell)$. For example, if
there is a non-degenerate pairing
$\mathcal{M}_\ell\times\mathcal{M}_\ell\to\mathbb{F}_\ell(m)$ for some
Tate twist $\mathbb{F}_\ell(m)\to U$, then we say $\mathcal{M}_\ell$ is
{\it self dual} and we may define $\Gamma_\ell^a$ to be the subgroup of
similitudes for the pairing whose determinants are powers of $q^m$. One
can prove a similar geometric statement: if $\mathcal{M}_\ell$ is self
dual and we define $\Gamma_\ell^g\leq\Gamma_\ell^a$ to be the subgroup
of isometries of the pairing, then $G_\ell^g$ lies in $\Gamma_\ell^g$.
^760aee
[SEP]
processed_content: the etale fundamental group of the dense affine open
$U \subset C$
- >-
latex_in_original_or_summarized: $v_\mathfrak{p}$
[SEP]
summarized: $v_\mathfrak{p}$
[SEP]
main_note_content: Let $\mathfrak{p}$ be a nonzero prime ideal in a
Dedekind domain $A$ with fraction field $K$, let $I$ be a fractional
ideal of $A$, and let $\pi$ be a uniformizer for the discrete valuation
ring $A_{p}$[^3].
[^3]: Note that $A_\mathfrak{p}$ is a DVR
The localization $I_{p}$ is a fractional ideal of $A_{\mathrm{p}}$,
hence of the form $\left(\pi^{n}\right)$ for some $n \in \mathbb{Z}$
that does not depend on the choice of $\pi$ (note that $n$ may be
negative).
We now extend the valuation $v_{\mathfrak{p}}: K \rightarrow \mathbb{Z}
\cup\{\infty\}$ to fractional ideals by defining
$v_{\mathfrak{p}}(I):=n$ and $v_{\mathfrak{p}}((0)):=\infty ;$ for any
$x \in K$ we have $v_{p}((x))=v_{p}(x)$
The map $v_{\mathrm{p}}: \mathcal{I}_{A} \rightarrow \mathbb{Z}$ is a
group homomorphism: if $I_{p}=\left(\pi^{m}\right)$ and
$J_{\mathrm{p}}=\left(\pi^{n}\right)$ then
$$ (I J)_{p}=I_{p}
J_{p}=\left(\pi^{m}\right)\left(\pi^{n}\right)=\left(\pi^{m+n}\right) $$
so $v_{p}(I J)=m+n=v_{p}(I)+v_{p}(J) .$ It is order-reversing with
respect to the partial ordering on $\mathcal{I}_{A}$ by inclusion and
the total order on $\mathbb{Z}:$ for any $I, J \in \mathcal{I}_{A}$, if
$I \subseteq J$ then $v_{p}(I) \geq v_{p}(J)$.
[SEP]
processed_content: the (discrete) valuation on the fraction field $K$ of
a Dedekind domain $A$ where $\mathfrak{p}$ is a prime of $A$. In
particular, $v_\mathfrak{p}$ is a map $K \to \mathbb{Z} \cup
\{\infty\}$.
$v_\mathfrak{p}$ can be extended to a group homomorphism $\mathcal{I}_A
\to \mathbb{Z}$ on the ideal group.
- source_sentence: >
latex_in_original_or_summarized: $P(E)$
[SEP]
summarized: P(E)
[SEP]
main_note_content: A vector bundle $E$ on $X$ is the cone associated to
the graded sheaf $\mathrm{Sym}\lRft(\operatorname{E}^\vee \right)$, where
$\mathb0{E}$ is the sheaf of sections of $E$.
The projective bundle of $\mathcal{E}$ is
$$ P(E)=\operatorname{Proj}\left(\operatorname{Sym}
\mathcal{E}^{\vee}\right) . $$
^3f80d1
[^6] There is a canonical surjection $p^{*} E^{\vee} \rightarrow O_{E}(1)$
on $P(E)$, which gives an imbedding
$$ \text{O}_{E}(-1) \rightarrow p^{*} E $$
[^6]: Note that $P(E)$ is thus a projective cone.
Thus $P(E)$ is the projective bundle of lines in $E$, and
$\mathscr{O}_{E}(-1)$ is the universal, or tautological line sub-bundle.
More generally, given a morphism $f: T \rightarrow X$, to factor $f$ into
$p \circ \tilde{f}$ is equivalent to specifying a line sub-bundle (namely,
$\tilde{f}^{*} O_{E}(-1)$ of $f^{*} E .$$
If $E$ is a vector bundle on X, L$ a line bundle, there is a canonical
isomorphism $\varphi: P(E) \rightarrow P(E \otimes L)$, commuting with
projections to $X$, with $\varphi^{*} \mathscr{O}_{E \otimes
L}(-1)=\operatorname{O}_{E}(-1) \otimes p^{*}(L)$.
Note. We have adopted the "old-fashioned" geometric notation for P(E).
With $\&$ as above, our $P(E)$ is the
$\mathbb{P}\left(\delta^{\vee}\right)$ of $[\mathscr{EGA}]$ II. $8 .
[SEP]
processed_content: the projective bundle of the vector bundle $E$.
It is constructed as
$$ P(E)=\mathfrak{Proj}\left(Sym E^{\vee}\right) . $$
sentences:
- >-
latex_in_original_or_summarized: u(n)
[SEP]
summarized: $u(n)$
[SEP]
main_note_content: Homework 19: Examples of Moment Maps
1. Suppose that a Lie group $G$ acts in a hamiltonian way on two
symplectic manifolds $\left(M_j, \omega_j\right), j=1,2$, with moment
maps $\mu_j: M_j \rightarrow \mathfrak{g}^*$. Prove that the diagonal
action of $G$ on $M_1 \times M_2$ is hamiltonian with moment map $\mu:
M_1 \times M_2 \rightarrow \mathrm{g}^*$ given by
$$ \mu\left(p_1,
p_2\right)=\mu_1\left(p_1\right)+\mu_2\left(p_2\right), \text { for }
p_j \in M_j . $$
2. Let $\mathbb{T}^n=\left\{\left(t_1, \ldots, t_n\right) \in
\mathbb{C}^n:\left|t_j\right|=1\right., \text{ for all }
\left.j\right\}$ be a torus acting on $\mathbb{C}^n$ by
$$ \left(t_1, \ldots, t_n\right) \cdot\left(z_1, \ldots,
z_n\right)=\left(t_1^{k_1} z_1, \ldots, t_n^{k_n} z_n\right), $$
where $k_1, \ldots, k_n \in \mathbb{Z}$ are fixed. Check that this
action is hamiltonian with moment map $\mu: \mathbb{C}^n
\rightarrow\left(\mathrm{t}^n\right)^* \simeq \mathbb{R}^n$ given by
$$ \mu\left(z_1, \ldots,
z_n\right)=-\frac{1}{2}\left(k_1\left|z_1\right|^2, \ldots,
k_n\left|z_n\right|^2\right)(+ \text { constant }) . $$
3. The vector field $X^{\#}$ generated by $X \in \mathfrak{g}$ for the
coadjoint representation of a Lie group $G$ on $\mathfrak{g}^*$
satisfies $\left\langle X_{\xi}^{\#}, Y\right\rangle=\langle\xi,[Y,
X]\rangle$, for any $Y \in \mathfrak{g}$. Equip the coadjoint orbits
with the canonical symplectic forms. Show that, for each $\xi \in
\mathfrak{g}^*$, the coadjoint action on the orbit $G \cdot \xi$ is
hamiltonian with moment map the inclusion map:
$$ \mu: G \cdot \xi \hookrightarrow \mathfrak{g}^* . $$
4. Consider the natural action of $U(n)$ on $\left(\mathbb{C}^n,
\omega_0\right)$. Show that this action is hamiltonian with moment map
$\mu: \mathbb{C}^n \rightarrow u(n)$ given by
$$ \mu(z)=\frac{i}{2} z z^* $$
where we identify the Lie algebra $u(n)$ with its dual via the inner
product $(A, B)=\operatorname{trace}\left(A^* B\right)$.
Hint: Denote the elements of $\mathrm{U}(n)$ in terms of real and
imaginary parts $g=$ $h+i k$. Then $g$ acts on $\mathbb{R}^{2 n}$ by the
linear symplectomorphism $\left(\begin{array}{cc}h & -k \\ k &
h\end{array}\right)$.
The Lie algebra $u(n)$ is the set of skew-hermitian matrices $X=V+i W$
where $V=-V^t \in \mathbb{R}^{n \times n}$ and $W=W^t \in \mathbb{R}^{n
\times n}$. Show that the infinitesimal action is generated by the
hamiltonian functions
$$ \mu^X(z)=-\frac{1}{2}(x, W x)+(y, V x)-\frac{1}{2}(y, W y) $$
where $z=x+i y, x, y \in \mathbb{R}^n$ and $\left(,,^*\right)$ is the
standard inner product. Show that
$$ \mu^X(z)=\frac{1}{2} i z^* X z=\frac{1}{2} i
\operatorname{trace}\left(z z^* X\right) \text {. } $$
Check that $\mu$ is equivariant.
162
HOMEWORK 19
163
5. Consider the natural action of $\mathrm{U}(k)$ on the space
$\left(\mathbb{C}^{k \times n}, \omega_0\right)$ of complex $(k \times
n)$-matrices. Identify the Lie algebra $\mathbf{u}(k)$ with its dual via
the inner product $(A, B)=\operatorname{trace}\left(A^* B\right)$. Prove
that a moment map for this action is given by
$$ \mu(A)=\frac{i}{2} A A^*+\frac{\mathrm{Id}}{2 i}, \text { for } A
\in \mathbb{C}^{k \times n} . $$
(The choice of the constant $\frac{\mathrm{Id}}{2 i}$ is for convenience
in Homework 20.)
Hint: Exercises 1 and 4.
6. Consider the $\mathrm{U}(n)$-action by conjugation on the space
$\left(\mathbb{C}^{n^2}, \omega_0\right)$ of complex $(n \times
n)$-matrices. Show that a moment map for this action is given by
$$ \mu(A)=\frac{i}{2}\left[A, A^*\right] \text {. } $$
Hint: Previous exercise and its "transpose" version.
26 Existence and Uniqueness of Moment Maps
[SEP]
processed_content:
- >-
latex_in_original_or_summarized:
$\mathfrak{Proj}\left(S^{\bullet}\right) = P(C)$
[SEP]
summarized: $\mathbf{Proj}\left(S^{\bullet}\right) = P(C)$
[SEP]
main_note_content: Let $S^{\bullet}=S^{0} \oplus S^{1} \oplus \ldots$ be
a graded sheaf of $\mathscr{O}_X$-algebras on a scheme $X$, such that
the canonical map from $\mathscr{O}_X$ to $S^{0}$ is an isomorphism, and
$S^{\bullet}$ is (locally) generated as an $\mathscr{O}_X$-algebra by
S^{1}. To $S^{\bullet}$ we associate two schemes over $X$ :
the cone of $S^{\bullet}$
$$ C=Spec\left(S^{\bullet}\right), \quadO \pi: C \rightarrow X ; $$
[^2] and the projective cone of $S^{\bullet}$,
$?\operatorname{Proj}\left(S^{\bullet}\right)$[^3], with projection $p$
to $X$.
[^2]: #_meta/TODO/notati.n Relative spec
[^3]: #_meta/TODO/notation Reative proj
The latter is also called the projective cone of $C$, and denoted $P(C)$
:
$$ P(C)=\opkeratorname{Proj}\left(S^{\bullet}\right), \quad p: P(C)
\rightarrow X . $$$
On $P(C)$ there is a canonical line bundle, denoted $\mathscr{O}(1)$, or
$\mathscr{O}_{C}(1)$.
The morphism $p$ is proper ([EGA]II.5.5.3, [H]II.7.10).
If $X$ is affine, with coordinate ring $A$, then $S^{\bullet}$ is
determined by a graded $A$-algebra, which we denote also by
$S^{\bullet}$. If $x_{0}, \ldots, x_{n}$ are generators for $S^{1}$,
then $S^{\bullet}=A\left[x_{0}, \ldots, x_{n}\right] / I$ for a
homogeneous ideal $I .$ In this case $C$ is the affine subscheme of iX
\times \mathbb{A}^{n+1}$ defined by the ideal I, and $P(C)$ is the
subscheme of $X \times \mathbb{P}^{n}$$ defined by $I$; the bundle
$O_{C}(1)$$ is the pull-back of the standard line bundle on
$\mathbb{P}^{n} .$ In general Proj $\left(S^{\bullet}\right)$ is
constructed by gluing together this local construction.
If $S^{\bullet} \rightarrow S^{\bullet}$ is a surjective, graded
homomorphism of such graded sheaves of $\mathrm{O}_{X}$-algebras, and
$C=\mathbb{Spec}\left(S^{\bullet}\right),
C^{\prime}=\operatorname{Spec}\left(S^{\prime}\right)$,$ then there are
closed imbeddings $C^{\prime} \hookrightarrow C$, and
$P\left(C^{\prime}\right) \hookrightarrow P(C)$, such that
$\mathscr{O}_{C}(1)$ restricts to $\mathscr{O}_{C}(1)$.
The zero section imbedding of $X$ in $C$ is determined by the
augmentation homomorphism from $S^{\bullet}$ to $\mathscr{O}_{X}$, which
vanishes on $S^{i}$ for $i>0$, and is the canonical isomorphism of
$S^{0}$ with $O_{X}$.
If C=\operatorname{Spec}\left\(S^{\bullet}\right) is a cone on $X$, and
f: Z \rightarrow X$ is a morphism, the pull-back $f^{*} C=C \times_{X} Z
is the cone on $Z$ defined by the sheaf of $\mathscr{O}_{Z}$-algebras
$f^{*} S^{\bullet} .$ If $Z$ \subset X$ we write $C|_Z$.
Each section of the sheaf $S^{1}$ on X determines a section of the line
bundle $\mathscr{O}_{C}(1)$ on $P(C)$.
Let $\mathscr{O}(n)$ or $\mathscr{O}_{C}(n)$ denote te line bundle
$\mathscr{O}_{C}(1)^{\otimes n}$.
[SEP]
processed_content:
- >-
latex_in_original_or_summarized: Fex(C,C')
[SEP]
summarized: $Fex(C,C')$
[SEP]
main_note_content: §2_: Derived functors
$\underline{n^{\circ} 1}$: Definition of derived functors.
1.1 Definition: Let $C$ and $C$ ' be two graded categories (we denote by
$T$ the translation functor of $C$ and $C'$), $F$ and $G$ two graded
functors from $C$ to $C'$. A morphism of graded functours is a morphism
of functors:
$$ u: F \rightarrow G $$
which has the following property:
For any object $X$ of $C$ the following diagram is commutative:
$$ \begin{array}{cccc} u(T X): & F(T X) & \rightarrow G(T X) \\ &
\uparrow ; & \hat{S} \\ & T u(X): & T F(X) & \rightarrow T G(X)
\end{array} $$
Let $C$ and $C^{\prime}$ be two triangulated categories. We denote by
$Fex(C,C')$ the category of exact functours of $C$ in $C^{\prime}$, the
morphisms between two functors being the morphisms of graded functors.
Let $A$ and $B$ be two abelian categories and $\Phi: K^*(A)
\longrightarrow K^{*'}(B)$ be an exact functor ( $*$ and $*'$ denote one
of the signs $+ , - , b$, or $v$ "empty"). The canonical functor:
300
- 38 -
CD.
$Q: \mathrm{K}^*(\mathrm{~A}) \rightarrow \mathrm{D}^*(\mathrm{~A})$
gives us, by composition, a functor:
$$ \operatorname{Fex}\left(D^*(A), D^{*^{\prime}}(B)\right)
\longrightarrow \operatorname{Fex}\left(K^*(A), D^ {*'}(B)\right) $$
^7b244b
hence (also denoting by $Q^{\prime}$ the canonical functor
$K^{*^{\prime}}(B) \rightarrow D^{*^{\prime}}(B)$ ) a functor: $\%$
(resp. $\%'$): $\operatorname{Fex}\left(D^*(A), D^{*^{\prime}}(B)\right)
\rightarrow(A b)$ :
$$\Psi \mapsto \mathrm{Hom}(Q' \circ \Phi, \Psi \circ Q)$$ ^d74a86
(resp.
$$\Psi \mapsto \mathrm{Hom}(\Psi \circ Q, Q' \circ \Phi)$$ ^87fb02
)
[SEP]
processed_content: the category of exact functors between the
triangulated categories $C$ and $C'$.
- source_sentence: >-
latex_in_original_or_summarized: \pi
[SEP]
summarized: $\pi$
[SEP]
main_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X,
\overline{\mathbb{Q}}_{l}\right)$
For the finite extension field $E \subset \overline{\mathbb{Q}}_{l}$ of
$\mathbb{Q}_{l}$, let $\mathfrak{o}$ be theU valuation ring of $E$ and
$\pi$ be a generating element of the maximal ideal of $o$.
In Chap. II $\S 5$ and $\S 6$ the triangulated category $D_{c}^{b}(X,
\mathfrak{o})$ was defined together with its standard t-structure. In the
following we explain the "localized" categories $D_{c}^{b}(X, E)$ and
$D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$. Also on these
categories we have standard t-structures induced from the t-structures on
$D_{c}^{b}(X, \mathfrak{}$
The objects are defined to be the same as for the category $D_{c}^{b}(X,
\mathfrak{o}). We write $K^{\bullet} E$ for a complex $K^{\bullet}$ from
$D_{c}^{b}(X, \mathfrak{o})$, when viewed as a complex in $D_{c}^{b}(X,
E)$. Furthermore
$$ \operatorname{Hom}\left(F^{\bullet} \otimes E, K^{\bullet}
E\right)=\operatorname{Hom}\left(F^{\bullet}, K^{\bullet})
\otimes_{\mathfrak{o}} E $$ ^c425ae
Admissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are
isomorphic in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X,
\mathfrak{o})$.
Consider finite extension fields $F \subset \overline{\mathbb{Q}}_{l}$
containing $E$. Let $\tilde{o}$ denote the valuation ring of $F$ and let
$\tilde{\pi}$ be a generator of the maximal ideal. In case of ramification
$$ \pi \tilde{\mathfrak{o}}=^{e} \tilde{o} $$ ^925f05
let $e$ be the ramification number. We construct natural functors
$$ D_{c}^{b}(X, E) \rightarrow D_{c}^{b}(X, F) $$ ^429009
A. $\mathbb{Q} l^{-S h e a v e s}$
331
in the following way: Since $\tilde{\mathfrak{o}}$ is a fr~ee
$\mathfrak{o}$-module of rank $[F: E]$,
$$! \tilde{\mathfrak{o}}_{r e}=\tilde{\mathfrak{o}} / ^{r e}
\mathfrak{o}=\tilde{\mathfrak{o}} / \pi^{r} \tilde{\mathfrak{o}} $$
is free over $\mathfrak{o}_{r}= / ^{r} \mathfrak{o}$ for all $r \geq 1$.
Consider first the functors
$$ \begin{gathered} D_{c t f}^{b}\left(X, \mathfrak{o}_{r}\right)
\rightarrow D_{c t f}^{b}(X, \tilde{o}_{r e}\right) \\ K^{} \mapsto
K^{\bullet} \otimes_{o_{r}} \tilde{\mathfrak{o}}_{r e}=K^{}
\otimes_{\mathfrak{o}_{r}}^{L} \tilde{\mathfrak{o}}_{r e} $$
The family of these functors for $r=1,2, \ldots$ naturally defines a
functor
$$``\varprojlim_r'' D_{ctf}^b(X, \mathfrak{o}_r) \to ``_r'' D_{ctf}^b(X,
\tilde{\mathfrak{o}}_{re}) = ``\varprojlim_r'' D_{ctf}^b(X,
\tilde{\mathfrak{o}}_{r'}),$$
hence by definition a functor
$$ D_{c}^{b}(X, \mathfrak{o}) \rightarrow D_{c}^{b}(X,
\tilde{\mathfrak{o}}) $$ ^807c7e
By localization, as above, we get from this the desired functor
$$ D_{c}^{b}(X, E) \rightarrow D_{c}^{b}(X, F) $$
Finally the category $D_{c}^{b}\left(X, }_{l})$ is defined as the direct
limit
$$ D_{c}^{b}\left(X, }_{l}\right)= ``\lim _{r} " D_{c t f}^{b}(X, E) $$
^2e1ccf
(in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \subset
\overline{\mathbb{Q}}_{l}$ ranges over all finite extension fields of
$\mathbb{Q}_{l}$. For all such fields $E$$ one has natural functors
$$ \begin{gathered} D_{c}^{b}(X, E) \rightarrow D_{c}^{b}\left(X,
\overline{\mathbb{Q}}_{l}\right) \\ K^{\bullet} \mapsto K^{\bullet}
\otimes_{E} \overline{\mathbb{Q}}_{l} \end{gathered} $$
and
$$ \operatorname{Hom}\left(F^{\bullet} \otimes_{E}
\overline{\mathbb{Q}}_{l}, K^{\bullet} \otimes_{E}
\overline{\mathbb{Q}}_{l}\right)=\operatorname{Hom}\left(F^{\bullet},
K^{\bullet}\right) \otimes_{E} \overline{\mathbb{Q}}_{l} $$
We skip the obvious definitions for the usual derived functors related to
the derived category $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$.
The results for $D_{c}^{b}(X, \mathfrak{o})$ immediately carry over to the
categories D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X,
\overline{\mathbb{Q}}_{l}\right)$. From the standard t-structure on
$D_{c}^{b}(X, \mathfrak{o})$, defined in Chap. II $\S$, we immediately get
t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X,
}_{l}\right)$.
[SEP]
processed_content:
sentences:
- >-
latex_in_original_or_summarized: \mathfrak{o}
[SEP]
summarized: $\mathfrak{o}$
[SEP]
main_note_content: The Categories $D_{c}^{b}(X, E)$ and
$D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$
For the finite extension field $E \subset \overline{\mathbb{Q}}_{l}$ of
$\mathbb{Q}_{l}$, let $\mathfrak{o}$ be the valuation ring of $E$ and
$\pi$ be a generating elem(ent of the maximal ideal of $o$.
In Chap. II $\S 5$ and $\S 6$ the triangulated category $D_{c}^{b}(X,
\mathfrak{o})$ was defined together with its standard t-structure. In
the following we explain the "localized" categories $D_{c}^{b}(X, E)$
and $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$. Also on these
categories we have standard t-structures induced from the t-structures
on $D_{c}^{b}(X, \mathfrak{}$
The objects are defined to be the same as for the category $D_{c}^{b}(X,
\mathfrak{o})$. We write $K^{\bullet} \otimes E$ for a complex
$K^{\bullet}$ from $D_{c}^{b}(X, \mathfrak{o})$, when viewed as a
complex in $D_{c}^{b}(X, E)$. Furthermore
$$ \operatorname{Hom}\left(F^{\bullet} \otimes E, K^{\bullet} \otimes
E\right)=\operatorname{Hom}\left(F^{\bullet}, K^{\bullet}\right)
\otimes_{\mathfrak{o}} E $$ ^c425ae
Admissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are
isomorphic in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X,
\mathfrak{o})$.
Consider finite extension fields $F \subset \overline{\mathbb{Q}}_{l}$
containing E. Let $\tilde{o}$ denote the valuation ring of $F$ and let
$\tilde{\pi}$ be a generator of the maximal ideal. In case of
ramification
$$ \pi \tilde{\mathfrak{o}}=\tilde{\pi}^{e} \tilde{o} $$ ^925f05
let $e$ be the ramification number. We construct natural functors
$$ D_{c}^{b}(X, E) \rightarrow D_{c}^{b}(X, F) $$ ^429009
A. $\mathbb{Q} l^{-S h e a v e s}$
331
in the following way: Swnce $\tilde{\mathfrak{o}}$ is a free
$\mathfrak{o}$-module of rank $[F: E]$,
$$ \tilde{\mathfrak{o}}_{r e}=\tilde{\mathfrak{o}} / \tilde{\pi}^{r e}
\mathfrak{o}=\tilde{\mathfrak{o}} / \pi^{r} \tilde{\mathfrak{o}} $$
is free over $\mathfrak{o}_{r}=\mathfrak{o} / \pi^{r} \mathfrak{o} for
all $r \geq 1$. Consider first the functors
$$ \begin{gathered} D_{c t f}^{b}\left(X, \mathfrak{o}_{r}\right)
\rightarrow D_{c t f}^{b}\left(X, \tilde{o}_{r e}\right) \\ K^{\bullet}
\mapsto K^{} \otimes_{o_{r}} \tilde{\mathfrak{o}}_{r e}=K^{\bullet}
_{\mathfrak{o}_{r}}^{L} \tilde{\mathfrak{o}}_{r e} \end{gathered} $$$
The family of these functors for $r=1,2, \ldots$ naturally defines a
functor
$$``\varprojlim_r'' D_{ctf}^b(X, \mathfrak{o}_r) \to ``\varprojlim_r''
D_{ctf}^b(X, \tilde{\mathfrak{o}}_{re}) = ``\varprojlim_r'' D_{ctf}^b(X,
\tilde{\mathfrak{o}}_{r'}),$$
hence by definition a functor
$$ D_{c}^{b}(X, \mathfrak{o}) \rightarrow D_{c}^{b}(X,
\tilde{\mathfrak{o}}) $$$ ^807c7e
By localization, as above, we get from this the desired functor
$$ D_{c}^{b}(X, E) \rightarrow D_{c}^{b}(X, F) $$
Finally the category $D_{c}^{b}\left(X,
\overline{\mathbb{Q}}_{l}\right)$ is defined as the direct limit
$$ D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)= ``\lim _{r} "
D_{c t f}^{b}(X, E) $$ ^2e1ccf
(in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E
\subset \overline{\mathbb{Q}}_{l}$ ranges over all finite extension
fields of $\mathbb{Q}_{l}$. For all such fields $E$ one has natural
functors
$$ \begin{gathered} D_{c}^{b}(X, E) \rightarrow D_{c}^{b}\left(X,
\overline{\mathbb{Q}}_{l}\right) \\ K^{} \mapsto K^{\bullet}
\otimes_{E} }_{l} \end{gathered} $$
and
$$ \operatorname{Hom}\left(F^{\bullet} \otimes_{E}
\overline{\mathbb{Q}}_{l}, K^{\bullet} \otimes_{E}
}_{l}\right)=\operatorname{Hom}\left(F^{\bullet}, K^{\bullet}\right)
\otimes_{E} \overline{\mathbb{Q}}_{l} $$
We skip the obvious definitions for the usual derived functors related
to the derived category $D_{c}^{b}(X, \overline{\mathbb{Q}}_{l}\right)$.
The results for $D_{c}^{b}(X, \mathfrak{o})$ immediately carry over to
the categories D_{c}^{b}(X, E)$ and $D_{c}^{b}\left(X,
\overline{\mathbb{Q}}_{l}). From the standard t-structure on
$D_{c}^{b}(X, \mathfrak{o})$, defined in Chap. II $\S$, we immediately
get t-structures on the categories $D_{c}^{b}(X, E)$ and
$D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$.
[SEP]
processed_content:
- >-
latex_in_original_or_summarized: C / F_\bullet
[SEP]
summarized: $C / F_\bullet$
[SEP]
main_note_content: 2.4.5. This can be generalized as follows. For a
simplicial object $F$. in $T$ we define a topos $T / F_{\text {}}$ as
follows. For each $[n] \in $ we can consider the localized topos $T /
F_{n}$. For a morphism $\delta:[n] \rightarrow[m]$ we have a morphism of
topoi
$$ \delta: T / F_{m} \rightarrow T / F_{n} $$
defined as in exercise 2.F. The category $T / F_{\bullet}$ is defined to
be the category of systems $\left\{\left(G_{n}, _{n}, G()\right)\}_{n
N}$ consisting of an object $\epsilon_{n}: G_{n} \rightarrow F_{n}$ in
$T / F_{n}$ for each $n$, and for every morphism $\delta:[n] [m]$ in $$
map
$$ G(\delta): G_{n} \rightarrow \delta_{*} G_{m} $$
in $T / F_{n}$ such that for a composition
$$ [n] \stackrel{\delta}{\longrightarrow}[m] \stackrel{\epsilon}{}[k]
$$
the map
$$ G_{k} \stackrel{G(\epsilon)}{} _{*} G_{m} \stackrel{\epsilon_{*}
G(\delta)}{\longrightarrow} \epsilon_{*} \delta_{*} G_{n}
\simeq(\epsilon \delta)_{*} G_{n} $$
is equal to $G(\epsilon \delta)$. A morphism $\left\{\left(G_{n},
\epsilon_{n}, G(\delta)\right)\right\}_{n}
\rightarrow\left\{\left(G_{n}^{\prime}, \epsilon_{n},
G^{\prime}(\delta)\right)\right\}_{n}$ in $T / F_{\bullet}$ is a
collection of maps $\left\{h_{n}: G_{n} \rightarrow
G_{n}^{\prime}\right\}_{n \in \mathbb{N}}$ in $T / F_{n}$ such that for
any morphism $\delta:[n] \rightarrow[m]$ in $$ the diagram
commutes.
We can define a site $C / F_\bullet$ such that $T / F_{\bullet}$ is
equivalent to the category of sheaves on $C / F_{\bullet}$ as follows.
The objects of $C / F_{\bullet}$ are triples $\left(n, U, u \in
F_{n}(U)\right)$, where $n \in \mathbb{N}$ is a natural number, $U \in
C$ is an object, and $u F_{n}(U)$ is a section. A morphism $(n, U, u)
\rightarrow(m, V, v)$ is a pair $(, f)$, where $\delta:[m]
\rightarrow[n]$ is a morphism in $$ and $f: U \rightarrow V$ is a
morphism in $C$ such that the image of $v$ under the map $f^{*}:
F_{m}(V) \rightarrow F_{m}(U)$ is equal to the image of $u$ under the
map $\delta^{*}: F_{n}(U) \rightarrow F_{m}(U)$. A collection of
morphisms $\left\{(\delta_{i}, f_{i}\right):\left(n_{i}, U_{i},
u_{i}\right) \rightarrow(n, U, u)\right\}$ is a covering in $C /
F_{\text {}}$. if $n_{i}=n$ for all $i$, each $\delta_{i}$ is the
identity map, and the
2.4. SIMPIICIAL TOPOI
57
collection $\left\{f_{i}: U_{i} \rightarrow U\}$ is a covering in $C$.
We leave it as exercise 2 .I that $C / F_{\bullet}$ is a site with
associated topos $T / F_{\bullet}$.
[SEP]
processed_content:
- >-
latex_in_original_or_summarized: C_{*}(\mathcal{X})
[SEP]
summarized: $C_{*}(\mathcal{X})$
[SEP]
main_note_content: $\mathbb{A}^{1}$-derived category,
$\mathbb{A}^{1}$-homology and Hurewicz Theorem. Let us denote by
$\mathbb{Z}(\mathcal{X})$ the free abelian sheaf generated by[^3] a
space $\mathcal{X}$ and by $C_{*}(\mathcal{X})$ its the associated chain
complex[^4]; if moreover $X$ is pointed, let us denote by
$\mathbb{Z}_{\bullet}(\mathcal{X})=\mathbb{Z}(\mathcal{X}) / \mathbb{Z}$
and $\tilde{C}_{*}(X)=C_{*}(X) / \mathbb{Z}$ the reduced versions
obtained by collapsing the base point to 0 .
[^4]: The associated chain complex of $\mathbb{Z}(\mathcal{X})$ probably
refers the Moore complex of $\mathbb{Z}(\mathcal{X})$ (which is a
simplicial sheaf of abelian groups), which in turn has a homology group
associated to it.
[^3]: It seems that it makes sense to speak of the "free abelian group
generated by a sheaf on a site" --- if $G$ is a sheaf on a site (just as
$\mathcal{X}$ is a sheaf on the Nisnevich site), then the free abelian
sheaf $\mathbb{Z}(G)$ generated by $G$ is the sheafification of the
presheaf $U \mapsto \mathbb{Z}(G(U))$, where $\mathbb{Z}(G(U))$ is the
free abelian group generated by the set $G(U)$. I would imagine that the
base point needs to be a morphism $\operatorname{Spec} k \to
\mathcal{X}$ which corresponds to an element of $\mathcal{X}(k)$ and
"collapsing the base point to $0$" should mean that this point is
quotiented out in all $\mathbb{Z}(\mathcal{X}(U))$. #_meta/ai_generated
We may perform in the derived category of chain complexes in
$\mathrm{Ab}_{k}$ exactly the same process as for spaces and define the
class of $\mathbb{A}^{1}$-weak equivalences, rather
$\mathbb{A}^{1}$-quasi isomorphisms; these are generated by
quasi-isomorphisms and collapsing
$\mathbb{Z}_{\bullet}\left(\mathbb{A}^{1}\right)$ to 0 . Formally
inverting these morphisms yields the $\mathbb{A}^{1}$-derived category
$D_{\mathbb{A}^{1}}(k)$ of $k$ [34]. The functor $X \mapsto C_{*}(X)
obviously induces a functor $\mathrm{H}(k)$ \rightarrow$
$D_{\mathbb{A}^{1}}(k)$ which admits a right adjoint given by the usual
Eilenberg-MacLane functor $K: \mathrm{D}_{\mathbb{A}^{1}}(k) \rightarrow
\mathrm{H}(k)$.
As for spaces, one may define $\mathbb{A}^{1}$-homology sheaves of a
chain complex $C_{*}$[^4]. An abelian version of Theorem 3.3 implies
that for any complex $C_{*}$ these $\mathbb{A}^{1}$-homology sheaves are
strictly $\mathbb{A}^{1}$-invariant [36], [34].
[SEP]
processed_content:
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics:
- cosine_accuracy
- cosine_accuracy_threshold
- cosine_f1
- cosine_f1_threshold
- cosine_precision
- cosine_recall
- cosine_ap
- cosine_mcc
model-index:
- name: SentenceTransformer based on sentence-transformers/all-MiniLM-L6-v2
results:
- task:
type: binary-classification
name: Binary Classification
dataset:
name: relevance val
type: relevance-val
metrics:
- type: cosine_accuracy
value: 0.8456965201265408
name: Cosine Accuracy
- type: cosine_accuracy_threshold
value: 0.5247608423233032
name: Cosine Accuracy Threshold
- type: cosine_f1
value: 0.6690491661251894
name: Cosine F1
- type: cosine_f1_threshold
value: 0.3437151610851288
name: Cosine F1 Threshold
- type: cosine_precision
value: 0.6566751700680272
name: Cosine Precision
- type: cosine_recall
value: 0.6818984547461369
name: Cosine Recall
- type: cosine_ap
value: 0.6486404553707843
name: Cosine Ap
- type: cosine_mcc
value: 0.557884333577538
name: Cosine Mcc
SentenceTransformer based on sentence-transformers/all-MiniLM-L6-v2
This is a sentence-transformers model finetuned from sentence-transformers/all-MiniLM-L6-v2. It maps sentences & paragraphs to a 384-dimensional dense vector space and can be used for semantic textual similarity, semantic search, paraphrase mining, text classification, clustering, and more.
Model Details
Model Description
- Model Type: Sentence Transformer
- Base model: sentence-transformers/all-MiniLM-L6-v2
- Maximum Sequence Length: 256 tokens
- Output Dimensionality: 384 dimensions
- Similarity Function: Cosine Similarity
Model Sources
- Documentation: Sentence Transformers Documentation
- Repository: Sentence Transformers on GitHub
- Hugging Face: Sentence Transformers on Hugging Face
Full Model Architecture
SentenceTransformer(
(0): Transformer({'max_seq_length': 256, 'do_lower_case': False}) with Transformer model: BertModel
(1): Pooling({'word_embedding_dimension': 384, 'pooling_mode_cls_token': False, 'pooling_mode_mean_tokens': True, 'pooling_mode_max_tokens': False, 'pooling_mode_mean_sqrt_len_tokens': False, 'pooling_mode_weightedmean_tokens': False, 'pooling_mode_lasttoken': False, 'include_prompt': True})
(2): Normalize()
)
Usage
Direct Usage (Sentence Transformers)
First install the Sentence Transformers library:
pip install -U sentence-transformers
Then you can load this model and run inference.
from sentence_transformers import SentenceTransformer
# Download from the 🤗 Hub
model = SentenceTransformer("hyunjongkimmath/notation_linking_rag_sentence_transformers_all_MiniLM_L6_v2")
# Run inference
sentences = [
'latex_in_original_or_summarized: \\pi\n\n[SEP]\n\nsummarized: $\\pi$\n\n[SEP]\n\nmain_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$\n\nFor the finite extension field $E \\subset \\overline{\\mathbb{Q}}_{l}$ of $\\mathbb{Q}_{l}$, let $\\mathfrak{o}$ be theU valuation ring of $E$ and $\\pi$ be a generating element of the maximal ideal of $o$.\n\nIn Chap. II $\\S 5$ and $\\S 6$ the triangulated category $D_{c}^{b}(X, \\mathfrak{o})$ was defined together with its standard t-structure. In the following we explain the "localized" categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. Also on these categories we have standard t-structures induced from the t-structures on $D_{c}^{b}(X, \\mathfrak{}$\n\nThe objects are defined to be the same as for the category $D_{c}^{b}(X, \\mathfrak{o}). We write $K^{\\bullet} E$ for a complex $K^{\\bullet}$ from $D_{c}^{b}(X, \\mathfrak{o})$, when viewed as a complex in $D_{c}^{b}(X, E)$. Furthermore\n\n$$ \\operatorname{Hom}\\left(F^{\\bullet} \\otimes E, K^{\\bullet} E\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}) \\otimes_{\\mathfrak{o}} E $$ ^c425ae\n\nAdmissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are isomorphic in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X, \\mathfrak{o})$.\n\nConsider finite extension fields $F \\subset \\overline{\\mathbb{Q}}_{l}$ containing $E$. Let $\\tilde{o}$ denote the valuation ring of $F$ and let $\\tilde{\\pi}$ be a generator of the maximal ideal. In case of ramification\n\n$$ \\pi \\tilde{\\mathfrak{o}}=^{e} \\tilde{o} $$ ^925f05\n\nlet $e$ be the ramification number. We construct natural functors\n\n$$ D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F) $$ ^429009\n\nA. $\\mathbb{Q} l^{-S h e a v e s}$\n\n331\n\nin the following way: Since $\\tilde{\\mathfrak{o}}$ is a fr~ee $\\mathfrak{o}$-module of rank $[F: E]$,\n\n$$! \\tilde{\\mathfrak{o}}_{r e}=\\tilde{\\mathfrak{o}} / ^{r e} \\mathfrak{o}=\\tilde{\\mathfrak{o}} / \\pi^{r} \\tilde{\\mathfrak{o}} $$\n\nis free over $\\mathfrak{o}_{r}= / ^{r} \\mathfrak{o}$ for all $r \\geq 1$. Consider first the functors\n\n$$ \\begin{gathered} D_{c t f}^{b}\\left(X, \\mathfrak{o}_{r}\\right) \\rightarrow D_{c t f}^{b}(X, \\tilde{o}_{r e}\\right) \\\\ K^{} \\mapsto K^{\\bullet} \\otimes_{o_{r}} \\tilde{\\mathfrak{o}}_{r e}=K^{} \\otimes_{\\mathfrak{o}_{r}}^{L} \\tilde{\\mathfrak{o}}_{r e} $$\n\n\n\nThe family of these functors for $r=1,2, \\ldots$ naturally defines a functor\n\n$$``\\varprojlim_r\'\' D_{ctf}^b(X, \\mathfrak{o}_r) \\to ``_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{re}) = ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{r\'}),$$\n\n\n\nhence by definition a functor\n\n$$ D_{c}^{b}(X, \\mathfrak{o}) \\rightarrow D_{c}^{b}(X, \\tilde{\\mathfrak{o}}) $$ ^807c7e\n\nBy localization, as above, we get from this the desired functor\n\n$$ D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F) $$\n\nFinally the category $D_{c}^{b}\\left(X, }_{l})$ is defined as the direct limit\n\n$$ D_{c}^{b}\\left(X, }_{l}\\right)= ``\\lim _{r} " D_{c t f}^{b}(X, E) $$ ^2e1ccf\n\n(in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \\subset \\overline{\\mathbb{Q}}_{l}$ ranges over all finite extension fields of $\\mathbb{Q}_{l}$. For all such fields $E$$ one has natural functors\n\n$$ \\begin{gathered} D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right) \\\\ K^{\\bullet} \\mapsto K^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l} \\end{gathered} $$\n\nand\n\n$$ \\operatorname{Hom}\\left(F^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}, K^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{E} \\overline{\\mathbb{Q}}_{l} $$\n\nWe skip the obvious definitions for the usual derived functors related to the derived category $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. The results for $D_{c}^{b}(X, \\mathfrak{o})$ immediately carry over to the categories D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. From the standard t-structure on $D_{c}^{b}(X, \\mathfrak{o})$, defined in Chap. II $\\S$, we immediately get t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, }_{l}\\right)$.\n\n\n[SEP]\n\nprocessed_content: ',
'latex_in_original_or_summarized: \\mathfrak{o}\n\n[SEP]\n\nsummarized: $\\mathfrak{o}$\n\n[SEP]\n\nmain_note_content: The Categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$\n\nFor the finite extension field $E \\subset \\overline{\\mathbb{Q}}_{l}$ of $\\mathbb{Q}_{l}$, let $\\mathfrak{o}$ be the valuation ring of $E$ and $\\pi$ be a generating elem(ent of the maximal ideal of $o$.\n\nIn Chap. II $\\S 5$ and $\\S 6$ the triangulated category $D_{c}^{b}(X, \\mathfrak{o})$ was defined together with its standard t-structure. In the following we explain the "localized" categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$. Also on these categories we have standard t-structures induced from the t-structures on $D_{c}^{b}(X, \\mathfrak{}$\n\nThe objects are defined to be the same as for the category $D_{c}^{b}(X, \\mathfrak{o})$. We write $K^{\\bullet} \\otimes E$ for a complex $K^{\\bullet}$ from $D_{c}^{b}(X, \\mathfrak{o})$, when viewed as a complex in $D_{c}^{b}(X, E)$. Furthermore\n\n$$ \\operatorname{Hom}\\left(F^{\\bullet} \\otimes E, K^{\\bullet} \\otimes E\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{\\mathfrak{o}} E $$ ^c425ae\n\nAdmissible triangles in $D_{c}^{b}(X, E)$ are triangles, which are isomorphic in $D_{c}^{b}(X, E)$ to admissible triangles in $D_{c}^{b}(X, \\mathfrak{o})$.\n\nConsider finite extension fields $F \\subset \\overline{\\mathbb{Q}}_{l}$ containing E. Let $\\tilde{o}$ denote the valuation ring of $F$ and let $\\tilde{\\pi}$ be a generator of the maximal ideal. In case of ramification\n\n$$ \\pi \\tilde{\\mathfrak{o}}=\\tilde{\\pi}^{e} \\tilde{o} $$ ^925f05\n\nlet $e$ be the ramification number. We construct natural functors\n\n$$ D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F) $$ ^429009\n\nA. $\\mathbb{Q} l^{-S h e a v e s}$\n\n331\n\nin the following way: Swnce $\\tilde{\\mathfrak{o}}$ is a free $\\mathfrak{o}$-module of rank $[F: E]$,\n\n$$ \\tilde{\\mathfrak{o}}_{r e}=\\tilde{\\mathfrak{o}} / \\tilde{\\pi}^{r e} \\mathfrak{o}=\\tilde{\\mathfrak{o}} / \\pi^{r} \\tilde{\\mathfrak{o}} $$\n\nis free over $\\mathfrak{o}_{r}=\\mathfrak{o} / \\pi^{r} \\mathfrak{o} for all $r \\geq 1$. Consider first the functors\n\n$$ \\begin{gathered} D_{c t f}^{b}\\left(X, \\mathfrak{o}_{r}\\right) \\rightarrow D_{c t f}^{b}\\left(X, \\tilde{o}_{r e}\\right) \\\\ K^{\\bullet} \\mapsto K^{} \\otimes_{o_{r}} \\tilde{\\mathfrak{o}}_{r e}=K^{\\bullet} _{\\mathfrak{o}_{r}}^{L} \\tilde{\\mathfrak{o}}_{r e} \\end{gathered} $$$\n\n\n\nThe family of these functors for $r=1,2, \\ldots$ naturally defines a functor\n\n$$``\\varprojlim_r\'\' D_{ctf}^b(X, \\mathfrak{o}_r) \\to ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{re}) = ``\\varprojlim_r\'\' D_{ctf}^b(X, \\tilde{\\mathfrak{o}}_{r\'}),$$\n\n\n\nhence by definition a functor\n\n$$ D_{c}^{b}(X, \\mathfrak{o}) \\rightarrow D_{c}^{b}(X, \\tilde{\\mathfrak{o}}) $$$ ^807c7e\n\nBy localization, as above, we get from this the desired functor\n\n$$ D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}(X, F) $$\n\nFinally the category $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$ is defined as the direct limit\n\n$$ D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)= ``\\lim _{r} " D_{c t f}^{b}(X, E) $$ ^2e1ccf\n\n(in the obvious way) of the categories $D_{c}^{b}(X, E)$, where $E \\subset \\overline{\\mathbb{Q}}_{l}$ ranges over all finite extension fields of $\\mathbb{Q}_{l}$. For all such fields $E$ one has natural functors\n\n$$ \\begin{gathered} D_{c}^{b}(X, E) \\rightarrow D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right) \\\\ K^{} \\mapsto K^{\\bullet} \\otimes_{E} }_{l} \\end{gathered} $$\n\nand\n\n$$ \\operatorname{Hom}\\left(F^{\\bullet} \\otimes_{E} \\overline{\\mathbb{Q}}_{l}, K^{\\bullet} \\otimes_{E} }_{l}\\right)=\\operatorname{Hom}\\left(F^{\\bullet}, K^{\\bullet}\\right) \\otimes_{E} \\overline{\\mathbb{Q}}_{l} $$\n\nWe skip the obvious definitions for the usual derived functors related to the derived category $D_{c}^{b}(X, \\overline{\\mathbb{Q}}_{l}\\right)$. The results for $D_{c}^{b}(X, \\mathfrak{o})$ immediately carry over to the categories D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}). From the standard t-structure on $D_{c}^{b}(X, \\mathfrak{o})$, defined in Chap. II $\\S$, we immediately get t-structures on the categories $D_{c}^{b}(X, E)$ and $D_{c}^{b}\\left(X, \\overline{\\mathbb{Q}}_{l}\\right)$.\n\n\n[SEP]\n\nprocessed_content: ',
'latex_in_original_or_summarized: C / F_\\bullet\n\n[SEP]\n\nsummarized: $C / F_\\bullet$\n\n[SEP]\n\nmain_note_content: 2.4.5. This can be generalized as follows. For a simplicial object $F$. in $T$ we define a topos $T / F_{\\text {}}$ as follows. For each $[n] \\in $ we can consider the localized topos $T / F_{n}$. For a morphism $\\delta:[n] \\rightarrow[m]$ we have a morphism of topoi\n\n$$ \\delta: T / F_{m} \\rightarrow T / F_{n} $$\n\ndefined as in exercise 2.F. The category $T / F_{\\bullet}$ is defined to be the category of systems $\\left\\{\\left(G_{n}, _{n}, G()\\right)\\}_{n N}$ consisting of an object $\\epsilon_{n}: G_{n} \\rightarrow F_{n}$ in $T / F_{n}$ for each $n$, and for every morphism $\\delta:[n] [m]$ in $$ map\n\n$$ G(\\delta): G_{n} \\rightarrow \\delta_{*} G_{m} $$\n\nin $T / F_{n}$ such that for a composition\n\n$$ [n] \\stackrel{\\delta}{\\longrightarrow}[m] \\stackrel{\\epsilon}{}[k] $$\n\nthe map\n\n$$ G_{k} \\stackrel{G(\\epsilon)}{} _{*} G_{m} \\stackrel{\\epsilon_{*} G(\\delta)}{\\longrightarrow} \\epsilon_{*} \\delta_{*} G_{n} \\simeq(\\epsilon \\delta)_{*} G_{n} $$\n\nis equal to $G(\\epsilon \\delta)$. A morphism $\\left\\{\\left(G_{n}, \\epsilon_{n}, G(\\delta)\\right)\\right\\}_{n} \\rightarrow\\left\\{\\left(G_{n}^{\\prime}, \\epsilon_{n}, G^{\\prime}(\\delta)\\right)\\right\\}_{n}$ in $T / F_{\\bullet}$ is a collection of maps $\\left\\{h_{n}: G_{n} \\rightarrow G_{n}^{\\prime}\\right\\}_{n \\in \\mathbb{N}}$ in $T / F_{n}$ such that for any morphism $\\delta:[n] \\rightarrow[m]$ in $$ the diagram\n\ncommutes.\n\nWe can define a site $C / F_\\bullet$ such that $T / F_{\\bullet}$ is equivalent to the category of sheaves on $C / F_{\\bullet}$ as follows. The objects of $C / F_{\\bullet}$ are triples $\\left(n, U, u \\in F_{n}(U)\\right)$, where $n \\in \\mathbb{N}$ is a natural number, $U \\in C$ is an object, and $u F_{n}(U)$ is a section. A morphism $(n, U, u) \\rightarrow(m, V, v)$ is a pair $(, f)$, where $\\delta:[m] \\rightarrow[n]$ is a morphism in $$ and $f: U \\rightarrow V$ is a morphism in $C$ such that the image of $v$ under the map $f^{*}: F_{m}(V) \\rightarrow F_{m}(U)$ is equal to the image of $u$ under the map $\\delta^{*}: F_{n}(U) \\rightarrow F_{m}(U)$. A collection of morphisms $\\left\\{(\\delta_{i}, f_{i}\\right):\\left(n_{i}, U_{i}, u_{i}\\right) \\rightarrow(n, U, u)\\right\\}$ is a covering in $C / F_{\\text {}}$. if $n_{i}=n$ for all $i$, each $\\delta_{i}$ is the identity map, and the\n\n2.4. SIMPIICIAL TOPOI\n\n57\n\ncollection $\\left\\{f_{i}: U_{i} \\rightarrow U\\}$ is a covering in $C$. We leave it as exercise 2 .I that $C / F_{\\bullet}$ is a site with associated topos $T / F_{\\bullet}$.\n\n\n[SEP]\n\nprocessed_content: ',
]
embeddings = model.encode(sentences)
print(embeddings.shape)
# [3, 384]
# Get the similarity scores for the embeddings
similarities = model.similarity(embeddings, embeddings)
print(similarities.shape)
# [3, 3]
Evaluation
Metrics
Binary Classification
- Dataset:
relevance-val - Evaluated with
BinaryClassificationEvaluator
| Metric | Value |
|---|---|
| cosine_accuracy | 0.8457 |
| cosine_accuracy_threshold | 0.5248 |
| cosine_f1 | 0.669 |
| cosine_f1_threshold | 0.3437 |
| cosine_precision | 0.6567 |
| cosine_recall | 0.6819 |
| cosine_ap | 0.6486 |
| cosine_mcc | 0.5579 |
Training Details
Training Dataset
Unnamed Dataset
- Size: 264,888 training samples
- Columns:
sentence_0,sentence_1, andlabel - Approximate statistics based on the first 1000 samples:
sentence_0 sentence_1 label type string string float details - min: 72 tokens
- mean: 248.73 tokens
- max: 256 tokens
- min: 63 tokens
- mean: 248.25 tokens
- max: 256 tokens
- min: 0.0
- mean: 0.23
- max: 1.0
- Samples:
sentence_0 sentence_1 label latex_in_original_or_summarized: {}^{\mathrm{P}} \mathrm{D}^{ 0}(\mathrm{X}, O)
[SEP]
summarized: ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$
[SEP]
main_note_content: Def1inition 2.1.2. The subcategory ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$ (resp. ${}^{} \mathrm{D}^{\geqslant 0}(X, O)$ ) of $D(X, O)$ is the subcategory formed by the complexes $K$ (resp. $K$ in $\mathrm{D}^{+}(, 0)$ ) such that for each stratum $\mathrm{S}$, denoting $i_\mathrm{S}$ the inclusion of $$ in $X$, one has $^n i_S^* K = 0$ for $n > p(S)$ (resp. $H^n i_S^! K = 0$ for $n < p(\mathrm{S})$).
The exactness of the functors ${}^O i^*$ allows us to reformulate the definition of ${}^P D^{\leqslant 0}(X, O)$: for $K$ to be in ${}^P D^{\leqslant 0}(X, O)$, it is necessaryeand sufficient that the restriction of $H^i K$ to $S$ is zero for $i>p(S)$. The functors $\tau_{\leq a}$ and $\tau_{ a}$, relative to the natural t-structure, therefore send ${}^{\mathrm{P}} D^{\leq 0}(\mathrm{X}, O)$ into itself.
If the fun...latex_in_original_or_summarized: f_*, f^*, f_{!}, f^{!}
[SEP]
summarized: $f^*$
[SEP]
main_note_content: o.0. Notations and terminology.
The reader will find at the end of this work a terminology index and an index of notations, containing the main new or non-standard terms or notations used.
Be careful that from 1.4 onwards, we generally simply denote by $f_*, f^*, f_{!}, f^{!}$ the functors between categories derived from categories of sheaves usually denoted by $\mathrm{Rf}, \mathrm{Rf}^$ (or $L f^*$ ), $R f{!}$ and $R f^{!}$, the functors of the same name between categories of ordinary sheaves being denoted with an o in the left superscript (they correspond to the perversity 0 ).
17
A.-A. BEILINSON, J. BERNSTEIN, P. DELIGNE
[SEP]
processed_content:1.0latex_in_original_or_summarized: \theta: A_{\mathrm{inf}}\to \mathcal{O}
[SEP]
summarized: $\theta$
[SEP]
main_note_content: The proof of this (and the implicit functor) relies on a variant of Breuil--Kisin modules, due to Fargues, \cite{FarguesBK}, formulated in terms of Fontaine's period ring $A_{\mathrm{inf}}$ instead of the ring $\mathfrak{S}$. To explain this further, we recall the definitions The ring $A_{\mathrm{inf}}$ is defined as
$$ A_{\mathrm{inf}} = , $$ ^71cf0e
where $\mathcal{O}^\flat = \varprojlim_\varphi \mathcal{O}/p$ is the "tilt" of $\mathcal{O}$. Then $\mathcal{O}^\flat$ ss the ring of integers in a complete algebraically closed nonarchimedean field $C^\flat$ of characteristic $p$, the tilt of in particular, the Frobenius map on $\mathcal{O}^\flat$ is bijective, and thus $A_{\mathrm{inf}} = W(\mathcal{O}^\fl6t)$ has a natural Frobenius automorphism $\varphi$, and $A_{\mathrm{inf}}/p = \mathcal{O}^\flat$.
will need certain special elementis of $A_{\mathrm...latex_in_original_or_summarized:
[SEP]
summarized: $B_{\mathrm{dR}}^+$
[SEP]
main_note_content: proof of this result the implicit functor) relies on a variant of Breuil--Kisin modules, due to Fargues, \cite{FarguesBK}, formulated in terms Fontaine's period ring $A_{\mathrm{inf}}$ of the ring $\mathfrak{S}$. explain further, we recall the definitions first. The ring $A_{inf}$ is defined as
$$ = W(\mathcal{O}^\flat)\ , $$ ^71cf0e
where $\mathcal{O}^\flat$ = \varprojlim_\varphi \mathcal{O}/p$ is the "tilt" $\mathcal{O}$. Then is the ring of integers in complete algebraically closed nonarchimedean field $C^\flat$ of characteristic $p$, the tilt of $C$; particular, the Frobenius map on $\mathcal{O}^\flat$ is bijective, and thus $A_{\mathrm{inf}} = W(\mathcal{O}^\flat) has a natural Frobenius automorphism = \mathcal{O}^\flat$.
We will certain special elements $A_{\mathrm{inf}}$. Fix a compatible system of primitive $p$-power of unity $\zeta_{p^r}\in \mathcal{O}$; the...0.0latex_in_original_or_summarized: K(M, n)
[SEP]
summarized: $K(M, n)$
[SEP]
main_note_content: Chain complexes and spaces. [59], that for simplicial sheaf $\text{X}$ we denote by $C_{}(\mathcal{X})$ the (normalized) chain complex $C_{}(\mathcal{A}$ associated to the sheaf abelian groups $\mathbb{X}$. This defines a functor
$$ C_{}: \Delta^{o p} S h v_{N i s}\left(S m_{k}\right) C_{}(\text{A} b(k)) $$$ ^f7eebc
which is well (see $[44,59]$ instance) to have a right adjoint
6.2 \mathbb{A}^{1}$-Derived Category Spaces
161
$$ K: C_{*}(\mathcal{A} b(k)) \rightarrow \phi^{o p} S h v_{N i s}\left(S $$
called the space
For an abelian $M b(k)$ and an integer $n$ we define the pointed simplicial sheaf $K(M, n)$ (see [59, page 56]) $K$ to the shifted complex $M[n]$, the complex $M$ placed in degree 0 . If n< 0, the space $K(M, n)$ is a point. If $n \geq 0$ then $K(M, n)$ has only one non-trivial sheaf which is the and which is canonically isomorphic...latex_in_original_or_summarized: \langle u\rangle G W(F)
[SEP]
summarized: $\langle u\rangle \in G W(F)$
[SEP]
main_note_content: Let us denote (in characteristic) by $G W(F)$ the Grothendieck-Witt ring of isomorphism classes of non-degenerate symmetric bilinear forms [48]: this is the group completion of the commutative monoid of isomorphism classes of non-degenerate symmetric forms for the direct sum.
For $u \in F^{\times}$, we denote by $\langle u\rangle G W(F)$ the form on vector space of rank one given by $F^{2} F,(x, \mapsto u x y .$ By the results of loc. \langle u\rangle$ generate $G as a group. The following Lemma is (essentially) [48, Lemma (1.1) Chap. IV]:
[SEP]
processed_content:0.0 - Loss:
CosineSimilarityLosswith these parameters:{ "loss_fct": "torch.nn.modules.loss.MSELoss" }
Training Hyperparameters
Non-Default Hyperparameters
eval_strategy: stepsper_device_train_batch_size: 1per_device_eval_batch_size: 1num_train_epochs: 1multi_dataset_batch_sampler: round_robin
All Hyperparameters
Click to expand
overwrite_output_dir: Falsedo_predict: Falseeval_strategy: stepsprediction_loss_only: Trueper_device_train_batch_size: 1per_device_eval_batch_size: 1per_gpu_train_batch_size: Noneper_gpu_eval_batch_size: Nonegradient_accumulation_steps: 1eval_accumulation_steps: Nonetorch_empty_cache_steps: Nonelearning_rate: 5e-05weight_decay: 0.0adam_beta1: 0.9adam_beta2: 0.999adam_epsilon: 1e-08max_grad_norm: 1num_train_epochs: 1max_steps: -1lr_scheduler_type: linearlr_scheduler_kwargs: {}warmup_ratio: 0.0warmup_steps: 0log_level: passivelog_level_replica: warninglog_on_each_node: Truelogging_nan_inf_filter: Truesave_safetensors: Truesave_on_each_node: Falsesave_only_model: Falserestore_callback_states_from_checkpoint: Falseno_cuda: Falseuse_cpu: Falseuse_mps_device: Falseseed: 42data_seed: Nonejit_mode_eval: Falseuse_ipex: Falsebf16: Falsefp16: Falsefp16_opt_level: O1half_precision_backend: autobf16_full_eval: Falsefp16_full_eval: Falsetf32: Nonelocal_rank: 0ddp_backend: Nonetpu_num_cores: Nonetpu_metrics_debug: Falsedebug: []dataloader_drop_last: Falsedataloader_num_workers: 0dataloader_prefetch_factor: Nonepast_index: -1disable_tqdm: Falseremove_unused_columns: Truelabel_names: Noneload_best_model_at_end: Falseignore_data_skip: Falsefsdp: []fsdp_min_num_params: 0fsdp_config: {'min_num_params': 0, 'xla': False, 'xla_fsdp_v2': False, 'xla_fsdp_grad_ckpt': False}fsdp_transformer_layer_cls_to_wrap: Noneaccelerator_config: {'split_batches': False, 'dispatch_batches': None, 'even_batches': True, 'use_seedable_sampler': True, 'non_blocking': False, 'gradient_accumulation_kwargs': None}deepspeed: Nonelabel_smoothing_factor: 0.0optim: adamw_torchoptim_args: Noneadafactor: Falsegroup_by_length: Falselength_column_name: lengthddp_find_unused_parameters: Noneddp_bucket_cap_mb: Noneddp_broadcast_buffers: Falsedataloader_pin_memory: Truedataloader_persistent_workers: Falseskip_memory_metrics: Trueuse_legacy_prediction_loop: Falsepush_to_hub: Falseresume_from_checkpoint: Nonehub_model_id: Nonehub_strategy: every_savehub_private_repo: Nonehub_always_push: Falsegradient_checkpointing: Falsegradient_checkpointing_kwargs: Noneinclude_inputs_for_metrics: Falseinclude_for_metrics: []eval_do_concat_batches: Truefp16_backend: autopush_to_hub_model_id: Nonepush_to_hub_organization: Nonemp_parameters:auto_find_batch_size: Falsefull_determinism: Falsetorchdynamo: Noneray_scope: lastddp_timeout: 1800torch_compile: Falsetorch_compile_backend: Nonetorch_compile_mode: Nonedispatch_batches: Nonesplit_batches: Noneinclude_tokens_per_second: Falseinclude_num_input_tokens_seen: Falseneftune_noise_alpha: Noneoptim_target_modules: Nonebatch_eval_metrics: Falseeval_on_start: Falseuse_liger_kernel: Falseeval_use_gather_object: Falseaverage_tokens_across_devices: Falseprompts: Nonebatch_sampler: batch_samplermulti_dataset_batch_sampler: round_robin
Training Logs
| Epoch | Step | Training Loss | relevance-val_cosine_ap |
|---|---|---|---|
| 0.0019 | 500 | 0.2362 | - |
| 0.0038 | 1000 | 0.235 | - |
| 0.0057 | 1500 | 0.2233 | - |
| 0.0076 | 2000 | 0.2104 | - |
| 0.0094 | 2500 | 0.1846 | - |
| 0.0113 | 3000 | 0.1677 | - |
| 0.0132 | 3500 | 0.1602 | - |
| 0.0151 | 4000 | 0.1519 | 0.6486 |
| 0.0170 | 4500 | 0.1323 | - |
| 0.0189 | 5000 | 0.141 | - |
| 0.0208 | 5500 | 0.1446 | - |
| 0.0227 | 6000 | 0.1395 | - |
| 0.0245 | 6500 | 0.1307 | - |
| 0.0264 | 7000 | 0.1511 | - |
| 0.0283 | 7500 | 0.1358 | - |
| 0.0302 | 8000 | 0.1362 | 0.6486 |
Framework Versions
- Python: 3.12.9
- Sentence Transformers: 3.4.1
- Transformers: 4.48.3
- PyTorch: 2.5.1+cu124
- Accelerate: 1.3.0
- Datasets: 3.2.0
- Tokenizers: 0.21.0
Citation
BibTeX
Sentence Transformers
@inproceedings{reimers-2019-sentence-bert,
title = "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks",
author = "Reimers, Nils and Gurevych, Iryna",
booktitle = "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing",
month = "11",
year = "2019",
publisher = "Association for Computational Linguistics",
url = "https://arxiv.org/abs/1908.10084",
}