# 10/22/2022 Lecture 19

## Finiteness

:::{.remark}
Last time: lots of results on $\pi_1^\et$.
We finished with an equivalence of categories 
\[
\Sh(X_\et; \zmod)^{\mathrm{lcc}}&\iso \Fin\modsleft{G}^{\disc}, \qquad G\da \pi_1^\et(X) \\
F_M &\mapsfrom M
.\]
First note that an lcc sheaf is locally constant, so locally representable, and the sheaf condition implies that it is globally representable.
The representing object is a finite etale $X\dash$scheme, since it's locally constant, represented by disjoint copies of $X$, and the restriction maps give gluing data.
This descent data is effective, using effective descent for finite maps.
A finite etale $X\dash$scheme is a $\pi_1^\et(X)\dash$set, and the module structure comes from Yoneda since $F$ itself has that structure.

In the other direction, one can explicitly write down the sheaf $F_M$.
Forgetting the module structure on $M$ yields a finite etale $\pi_1^\et(X)\dash$set, which is the same as a finite etale $X\dash$scheme $Y_M$, so take the functor it represents $F_M \da \Hom(\wait, Y_M)$.
This is lcc since it's represented by a finite etale $X\dash$scheme, which is locally trivial in the etale topology -- e.g. take the Galois closure of a covering and pullback to split it.
Moreover this is a sheaf of abelian groups since a module has an underlying abelian group and the constructions are functorial.

:::

:::{.proposition title="?"}
Given this data, the proposition was that there is a map $H^i_\cts(\pi_1^\et, M)\to H^i(X_\et; F_M)$ inducing an isomorphism for $i=0, 1$.
:::

:::{.proof title="?"}
There is a morphism of sites $X_\et \to X_{\fet}$ induced by inclusion, where the latter has a Grothendieck topology where covers are surjections.
The claim is that there is an equivalence of categories 
\[
X_{\fet} \isovia{\pi} \Fin\gset{\pi_1^\et(X)}^\cts
.\]
A nontrivial fact about group cohomology is that 
\[
\Sh( \Fin\gset{\pi_1^\et(X)}^\cts, \Fin\zmod) \iso \Fin\mods{\pi_1^\et(X)}^{\cts,\disc}
.\]
The proof of this is morally the same as what one uses to show that sheaves on $\spec k_\et$ are the same as Galois modules.[^ctsanima]

There is a map 
\[
\pi^*: H^i(X_{\fet}; M)\to H^i(X_\et; \pi^* M)
,\]
which we argued could be shown to be an isomorphism in low degrees using the Leray spectral sequence.
Computing $\RR^i \pi_* \pi^* M$, one shows this is zero for $i=1$.

[^ctsanima]: 
Sheaves on the category of arbitrary continuous $\pi_1\dash$sets (not just finite ones, e.g. profinite sets) are closely related to the category of *anima*.

:::

:::{.corollary title="?"}
We can reduce computations in the follow way: 
\[
H^1(\pi_1(X, \bar x); M) \iso H^1(X_\et; F_M)
,\]
where the RHS is something that can sometimes be computed.
:::

:::{.remark}
The statement that $\pi_1 \AA^1$ was not topologically finitely-generated was conditional on this corollary.
Taking $M$ to be a trivial $\pi_1^\et\dash$module, $H^1$ consists of group homomorphisms, so it's not hard to compute maps from $\pi_1$ into other groups.
The general computational technique in this area is to use devissage repeatedly until one lands in a very simple and concrete situation.
:::

:::{.theorem title="Finiteness"}
Let $X\in\Var\slice k$ with $\ksep = k$ and suppose $F \in \Sh(X_\et)$ is constructible.
Then if either

1. $X$ is proper, or 
2. The stalks of $F$ all have order coprime to $\characteristic k$, 

then $H^r(X_\et; F)$ is finite.

:::

:::{.proof title="?"}
Note that the hypotheses are necessary: consider $\AA^1$ and $C_p$.
The first part immediately follows from proper base change.

For (2), we induct on dimension.

**Step 0**: We know it's true in dimensions 0 and 1, since we know it for smooth curves and there's only a small argument to extend to singular curves by reducing to the normalization.

**Step 1**: We'll make the simplifying assumption that $X$ is smooth -- we don't have resolutions of singularities in positive characteristic, but the alterations theorem makes reduction to the smooth case easier.


**Step 2**: Further assume $X = U$ fits into an elementary fibration $(U, Y, Z)$ over $S$, so $Y\to S$ is smooth of relative dimension 1, $Z\to S$ is finite étale, and $U\injects Y$ is fiberwise dense.

\begin{tikzcd}
	U & Y & Z \\
	& S
	\arrow["h", from=1-2, to=2-2]
	\arrow["f"', from=1-1, to=2-2]
	\arrow["i", from=1-1, to=1-2]
	\arrow[from=1-3, to=1-2]
	\arrow[from=1-3, to=2-2]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJVIl0sWzEsMCwiWSJdLFsyLDAsIloiXSxbMSwxLCJTIl0sWzEsMywiaCJdLFswLDMsImYiLDJdLFswLDEsImkiXSxbMiwxXSxbMiwzXV0=)


The idea is to cover $X$ by such $U$, and use the Cech-to-derived spectral sequence to compute the cohomology of $X$ using that of $U$.
The obstructions come from the fact that intersections might not fit into elementary fibrations, but the idea is that the intersections $U_i \fiberprod{X} U_j$ can themselves be covered by elementary fibrations, and one can use hypercovers and argue by pure homological algebra.

**Step 3**: Use dévissage to reduce to the case where $F$ is lcc using a Gysin argument.

**Step 4**: Use the Leray spectral sequence:
\[
H^i(S; \RR^i f_* F)\abuts H^{i+j}(U, F)
,\]
which we're trying to show is finite.
By induction on dimension, it's ETS that $\RR^j f_*F$ is constructible for all $j \geq 0$.

**Step 5**: Computing $\RR^j f_* F$.
Write $f = h \circ i$, and use the composition spectral sequence: 
\[
\RR^s h_* \RR^t i_* F \abuts \RR^{s+t} f_* F
.\]
Observe that $h$ is proper by definition, so to show that the RHS are constructible sheaves, it's ETS that $\RR i_* F$ are constructible by proper base change.

**Step 6**: It's ETS that in an elementary fibration, the derived pushforward along $U\injects Y$ of an lc sheaf is constructible.

Note: what happens when $F$ is constant, so $F = i_* \ul L$?
Then $i_* F = \ul{L}_Y$ is constant on $Y$, and $\RR^1 i_* \ul L = j_* \ul L(m)$ is the pushforward along $j$ of a twisted constant sheaf for some $m$.
This is supported along $j$, and one uses purity to understand what happens along $Z$.
Finally, $\RR^{i \geq 2} i_* \ul L = 0$, again using purity.
The hypotheses of coprimality of order and characteristic is used in computing $\RR j_*$.
:::

## Sheaves of $\ZZladic\dash$modules

:::{.remark}
The ultimate goal: we want to count points on varieties over finite fields, which we'll do by taking traces of Frobenius on cohomology.
In order to have those counts be well-defined integers and not just $\mod \ell^{10^{23}}$, the traces need to live in a ring in which no nonzero integer is zero.
If our cohomology of sheaves is finite, one has no chance of counting anything except modulo a large integer!
Sheaves of $\ZZladic\dash$modules will be the formalism used to compute in characteristic zero.
:::

:::{.definition title="Sheaves of $\ZZladic\dash$modules"}
A sequence $(M_n, f_{n+1}: M_{n+1}\to M_n)$ is a **sheaf of $\ZZladic\dash$modules** iff 

1. Each $M_n$ is a constructible sheaf of $C_{\ell^n}$modules, and 
2. Each $f_{n+1}$ induces an isomorphism $M^{n+1}/\ell^{n+1} M^{n+1} \iso M_n$.

:::

:::{.remark}
Motivation: an $\ell\dash$complete $\ZZladic\dash$module $N$, one has $N = \cocolim_n N\ell^n N$ and thus $N$ is determined by the data of $N/\ell^n N$ along with the transition maps.
So this definition is meant to capture being a complete sheaf of $\ZZladic\dash$modules, although the topology of $\ZZladic$ intervenes in this picture in a way that was not originally known how to handle (although we do know now). 
:::

:::{.example title="?"}
Let $M_n \da \ul{C_{\ell^n}}$ with $f_{n+1}: \ul{C_{\ell^{n+1}}}\to \ul{C_{\ell^n}}$ the obvious quotient.
Note that the stalks are all $C_{\ell^n}$, and taking the inverse limit yields a free $\ZZladic\dash$module, namely $\ZZladic$.
:::

:::{.definition title="Flat $\ZZladic\dash$modules"}
A sheaf of $\ZZladic\dash$modules is **flat** if there is a SES 
\[
M_s \injectsvia{\ell^n} M_{n+s}\surjects M_n
.\]
Motivation: this exactness characterizes flat $\ell\dash$complete $\ZZladic\dash$modules.
:::

:::{.definition title="Cohomology of $\ZZladic\dash$modules"}
Let $(M, f_n)$ be a sheaf of $\ZZladic\dash$modules, then define its **cohomology** as 
\[
H^r(X_\et; M) &\da \cocolim_n H^r(X_\et; M_n) \\
H^r_c(X_\et; M) &\da \cocolim_n H_c^r(X_\et; M_n)
,\]
where the inverse system in cohomology is induced by the transition maps.
:::

:::{.example title="?"}
Let $X$ be a smooth curve of genus $g$ over a field $k=\ksep$ with $\characteristic k \neq \ell$.
Then 
\[
H^i(X_\et; \ZZladic) \da \cocolim_n H^i(X_\et, C_{\ell^n})
= \ZZladic t^0 + T_\ell \Jac(X)(-1) t^1 + \ZZladic(-1) t^2
,\]
where we can note that $T_\ell \Jac(X) (-1) \cong \ZZladic^2$.
:::

:::{.theorem title="?"}
If $M$ is a flat sheaf of $\ZZladic\dash$modules on $X\in \Var\slice k$ with $k=\ksep$, if $X$ is proper or $\ell \neq \characteristic k$ then 

1. $H^r(X_\et; \ZZladic)\in \mods{\ZZladic}^\fg$,
2. There is a LES 
\[
H^{r-1}(X_\et; M_n) \to H^r(X_\et; M) \mapsvia{\ell^n} H^r(X_\et; M)\to H^r(X_\et; M_n)
.\]
Thus one can relate the inverse limit to the pieces, and one should think of this as the LES associated to the "SES"
\[
M \injectsvia{\ell^n} M \surjects M_n
.\]
:::

:::{.proof title="?"}
Reduce to finite quotients to reduce to the previous finiteness theorem, and build the LES here out of the LESs in cohomology arising from $M_s \injectsvia{\ell^n} M_{n+s} \surjects M_n$ by taking the inverse limit over $s\to \infty$.
Note that inverse limits are not exact, so one needs a short argument.
:::

:::{.definition title="Locally constant $\ZZladic\dash$sheaves"}
A $\ZZladic\dash$sheaf $(M, f_{n})$ is **locally constant** if each $M_n$ is locally constant (equivalently lcc here), and **lisse**[^lisse] if locally constant and flat.
One should think of lisse sheaves as local systems.

[^lisse]: 
French for smooth
:::

:::{.warnings}
$(M, f_{n})$ being locally constant need not imply that there exists a cover of $X$ on which it is constant.
This is essentially because one may need to continually refined the cover while passing to the inverse limit.
:::

:::{.remark}
There are representations of $\pi_1^\et$ associated to lc $\ZZladic\dash$sheaves:  letting $(M, f_n)$ be a lc $\ZZladic\dash$sheaf, then there is a continuous map
\[
\rho: \pi_1^\et(X)\to \Aut(M_{n, \bar x})
.\]
This follows from $M_n$ be a finite continuous $\pi_1^\et(X)\dash$module.
Moreover these live in a tower where $\Aut(M_{n+1}, \bar x)\to \Aut(M_n, \bar x)$, thus locally constant $\ZZladic\dash$sheaves are the same as continuous representations of $\pi_1^\et(X)$ on finitely-generated flat (and thus free) $\ZZladic\dash$modules.
A locally constant $\ZZladic\dash$sheaf is actually locally constant, i.e. trivialized by a cover, if the associated representation has finite image (which is very rare).
:::

:::{.remark}
There is a category of $\QQladic\dash$sheaves, defined as the localization of $\ZZladic\dash$sheaves at the Serre subcategory of sheaves where all $M_n$ have bounded order.
The objects will be the same as $\ZZladic\dash$sheaves, while a morphism $M\to N$ is a span $\catspan{M}{M'}{N}$ whose left leg $f$ has finite kernel and cokernel and $M'\to N$ is a levelwise morphism.
Given such a $\QQladic\dash$sheaf $M$, we define 
\[
H^i(X_\et; M) &\da \cocolim_n H^i(X_\et; M_n)\tensor \QQladic \\
H^i_c(X_\et; M) &\da \cocolim_n H^i(X_\et; M_n)\tensor \QQladic
.\]
The point is that spans will induce isomorphisms in these cohomology groups, so we formally invert them.
Note that this is similar to the isogeny category of abelian varieties, where morphisms with finite kernel are inverted.
:::

:::{.claim}
Given a $\QQladic\dash$sheaf whose underlying $\ZZladic\dash$sheaf is locally constant, we get a representation 
\[
\rho: \pi_1^\et(X, \bar x) \to \GL_n(\QQladic)
,\]
and this induces an equivalence of categories.
Given a $\QQladic\dash$representation $\rho$ of $\pi_1^\et(X)$, $\rho$ is conjugate to a representation into $\GL_n(\ZZladic)$ using that $\pi_1^\et$ is a profinite group.

:::

:::{.proof title="of claim"}
One proof uses that $\pi_1^\et$ is compact, thus has compact image, and the maximal compact subgroups of $\GL_n(\QQladic)$ are classified and all conjugate to $\GL_n(\ZZladic)$.

This classification: take a lattice, e.g. $\ZZladic^n \subseteq \QQladic^n$, then it is a topological fact for profinite groups that $\Stab_{\pi_1^\et}(\ZZladic^n)$ is open and hence finite index.
This is because one can pick a generator of $\ZZladic^n$, then the elements that send it back into $\ZZladic^n$ give an open condition, and there are finitely many generators so one can take a finite intersection of open subgroups which is nonempty since it contains the identity.
One can now average: 
\[
\sum_{g\in \pi_1^\et /\Stab_{\pi_1^\et}(\ZZladic^n) } g\ZZladic^n
,\]
which yields $\pi_1^\et\dash$stable element which is finitely-generated since this is a finite sum, and thus isomorphic to $\ZZladic^n$ by a commutative algebra argument -- any finitely-generated torsionfree $\ZZladic\dash$module is free.

So picking a basis of the lattice gives a conjugation into $\GL_n(\ZZladic)$, and a lisse $\ZZladic\dash$sheaf is the same as a representation into $\GL_n(\ZZladic)$.
:::