Model: hyunjongkimmath/notation_linking_rag_sentence_transformers_all_MiniLM_L6_v2
Author: hyunjongkimmath
Pipeline: sentence-similarity
Library: sentence-transformers
Tags: sentence-transformers, safetensors, bert, sentence-similarity, feature-extraction, generated_from_trainer, dataset_size:264888, loss:CosineSimilarityLoss, arxiv:1908.10084, base_model:sentence-transformers/all-MiniLM-L6-v2, base_model:finetune:sentence-transformers/all-MiniLM-L6-v2, model-index, text-embeddings-inference, endpoints_compatible, region:us
Base models (finetune): id: sentence-transformers/all-MiniLM-L6-v2
Card 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 / 

[SEP]

summarized: $j_0$

[SEP]

main_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.

(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 $\mathcal{P}$.

(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 $_c\left(_m / , M\right)$.

(3) Denoting by

$$  j_0: \mathbb{G}_m / \bar{k} \subset \mathbb{A}^1 / \bar{k}  $$ ^212b11

1. OVERVIEW

11

the inclusion, the construction

$$  M \mapsto H^0\left(\mathbb{A}^1 / \bar{k}, j_{0!} M\right)  $$ ^425e70

is 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.

(4) For any finite extension field $E / k$, and any multiplicative character $\rho$ of $E^{\times}$, the construction

$$  M \mapsto H^0\left(\mathbb{A}^1 / \bar{k}, j_{0!}\left(M \otimes \mathcal{L}_\rho\right)\right)  $$ ^f07855

is 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.

Now 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 $$ (any subquotient of $$ ).

[^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 $\omega$.

^370dc9

And 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).

For $\rho$ a multiplicative character of a finite extension field $E / k$, we have the fibre functor $\omega_\rho$ defined by

$$  M \mapsto H^0\left(\mathbb{A}^1 / \bar{k}, j_{!}\left(M  \mathcal{L}_\rho\right)\right)  $$

on $\langle N\rangle_{\text {arith }}$. 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

12

1. OVERVIEW

$\omega_0$, of fibre functor, and define

$$  G_{\text {arith }, N}:=G_{\text {arith }, N, \omega_0}, \quad G_{\text {geom }, N}:=G_{\text {geom }, N, \omega_0} .  $$

Then the element $Frob_{E, \rho}$ in the group $G_{\text {arith }, N, \omega_\rho}$ still makes sense as a conjugacy class in the group $G_{\text {arith }, N}$.

Let us say that a multiplicative character $\rho$ of some finite extension field $E / k$ is good for $N$ if, for

$$  j: \mathbb{G}_m / \bar{k} \subset \mathbb{P}^1 / \bar{k}  $$

the inclusion, the canonical "forget supports" map

$$  R j_1\left(N \otimes L_\right)  R j_{\star}\left(N \otimes _\rho\right)  $$

is an isomorphism. If $\rho$ is good for $N$, then the natural "forget supports" maps

$$  H_c^0\left(\mathbb{G}_m / , N \otimes \mathcal{L}_\rho\right)=H_c^0\left(\mathbb{A}^1 / \bar{k}, j_{0!}(N \otimes \mathcal{L}_\rho)\right) \rightarrow H^0\left(\mathbb{A}^1 / \bar{k}, j_{0!}\left(N \otimes L_\rho\right)\right),  $$

together with the restriction map

$$  H^0\left(^1 / \bar{k}, j_{0!}(N \otimes \mathcal{L}_\rho\right))  H^0\left(\mathbb{G}_m , N  _\rho\right),  $$

are all isomorphisms. Moreover, as $N$ is $$-pure of weight zero, each of these groups is $t$-pure of weight zero.

Conversely, 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

$$  H_c^0\left(\mathbb{G}_m / \bar{k}, N \otimes \mathcal{L}_\rho\right)  _\rho(N):=H^0\left(\mathbb{A}^1 / \bar{k}, j_{0!}\left(N \otimes \mathcal{L}_\rho\right)) .  $$

This 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

$$  H_c^0\left(\mathbb{G}_m / \bar{k}, M \otimes \mathcal{L}_\rho\right) \cong \omega_\rho(M) \text {. }  $$

Recall 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

1. OVERVIEW

13

either 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 $N$.

Recall [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

[^7]: Apparently, "completely reducible" is a synonym for "semisimple", cf. https://math.stackexchange.com/questions/334178/definition-completely-reducible-group-representation

[^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 $0$:
	1. $G$ is reductive
	2. every finite-dimensional representation of $G$ is semisimple
	3. some faithful finite dimensional representation of $G$ is semisimple.
	See also the proof of forey_fresan_kowalski_aftff_3.18 Corollary, which uses this theorem.

Let 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

$\operatorname{Trace}\left(\Lambda_K\left(\theta_{E, \rho}\right)\right)=\left(\Lambda\left(\operatorname{Frob} _{E, }^{s s})\right)=\operatorname{Trace}\left(\Lambda\left(\operatorname{Frob} \theta_{E, \rho}\right)\right)$. ^d42132

With these preliminaries out of the way, we can state the main theorem.


[SEP]

processed_content: the inclusion 

$$  j_0: \mathbb{G}_m / \bar{k}  \mathbb{A}^1 / \bar{k}  $$

The construction

$$  M \mapsto H^0\left(\mathbb{A}^1 / \bar{k}, j_{0!} M\right)  $$

is 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 $\mathscr{O}_{C}(1)^{\otimes n}$.


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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

)


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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 $D_{c}^{b}\left(X, }_{l}\right)$.


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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 $D_{c}^{b}\left(X, \overline{\mathbb{Q}}_{l}\right)$.


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summarized: $C / F_\bullet$

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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 $T / F_{\bullet}$.


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summarized: $C_{*}(\mathcal{X})$

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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]. 


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Likes: 0 | All-time downloads: 587
Created: 2025-02-26 22:19:51