Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ?

2010-04-29T16:07:15Z

Let me recall two theorems :

Let $K$ be a field, $\overline{K}$ be a separable closure of $K$ with absolute Galois group $G_K:=Gal(\overline{K}/K)$, and let $\ell$ be a prime that is different from $char(K)$. Let $X$ be a $K$-scheme that is separated and of finite type. The $\ell$-adic cohomology groups with proper support $H^i_c(X_{\overline{K}},\mathbb{Q}_{\ell})$.

They are finite dimensional $\mathbb{Q}_{\ell}$ vector spaces and are zero for $i>2\cdot \text{dim}(X)$ and $G_K$ acts (via monodromy, as the fundamental group) continuously on them, so that for each $g\in G_K$, the trace $$\text{Tr}(g,H^\ast_c(X_{\overline{K}},\mathbb{Q}_{\ell}))=\sum(-1)^i\text{Tr}(g,H^i_c(X_{\overline{K}},\mathbb{Q}_{\ell}))$$ is defined. This $\ell$-adic number is in fact an $\ell$-adic integer.

Write $K$ as the inductive limit of its $\mathbb{Z}$-sub-algebras of finite type. A model of $X/K$ over such a subring $R$ is a scheme $\mathcal{X}$ that is separated and of finite type over $S=Spec(R)$ such that $\mathcal{X}\times_{S} Spec(K)= X$. A model of $X/K$ is essentially unique ``up to shrinking'': two models $\mathcal{X}_1/R_1$ and $\mathcal{X}_2/R_2$ become isomorphic over some $S=Spec(R)$ with $R\supseteq R_1,R_2$ [EGA4,section 8].

Theorem 1 [Serre, 2004] Under the above notations and assumptions, for a positive integer $n$, the following conditions are equivalent:

There exists a model $\mathcal{X}/S$ of $X/K$ having the following property: for all points $s=Spec(k')\to S$ with value in a finite field $k'$ of characteristic different from $\ell$, we have $$|\mathcal{X}(s)|\equiv 0\text{ }( \text{ mod } \ell^n)$$

For all $g\in G_K$, we have $$\text{Tr}(g,H_c^\ast(X_{\overline{K}},\mathbb{Q}_{\ell}))\equiv 0\text{ }( \text{ mod } \ell^n)$$

This result is published in Illusie's note Miscellany on traces in $\ell$-adic cohomology from 2005.

On the other hand, let me recall Saito's approach (1987) to stable reduction of curves :

let $C$ be a curve proper and smooth over $K=Frac(R)$, $R$ a complete DVR with algebraically closed residual field $k(R)$ of characteristic $p>0$.

Theorem 2 [Stable reduction] Let $C$ be a geometrically irreducible proper smooth curve over $K$ of genus $g\geq 2$. Then there exists a finite separable extension $K'/K$ such that $C_{K'} = C\times_K K'$ admits a stable model over the integral closure $R'$ of $R$ in $K'$. I.e., $C_{K}$ is the generic fibre of some $C_m/R$ which is proper, flat, separated of finite type of relative dimension 1 whose geometric fibres are stable curves (i.e., they are reduced, connected, with only ordinary double points as singularities, and each connected component intersects the others in at least 3 points).

Saito's proof essentially shows that whether a curve admits a stable model (i.e., $K'=K$) is encoded by the representation $H^1(C_{\overline{K}},\mathbb{Q}_{\ell})$ of $Gal(\overline{K}/K)$ :

Theorem [Saito] Let $C$ be a geometrically irreducible proper smooth curve over $K$ of genus $g\geq 2$. The following conditions are equivalent :

$C$ admits a stable model

the action of inertia $I_K$ on $H^1(C_{\overline{K}},\mathbb{Q}_{\ell})$ is unipotent.

I refer you to Abbes' article in Courbes semi-stables et groupe fondamental en géométrie algébrique, Birkhäuser (1998) for a discussion of Saito's proof.

Question Do you know of other theorems of a similar flavor ? Are there general remarks on why such results might not be surprising ?

Answers I'm looking for would : state other theorems of a similar flavor, or explain why such theorems shouldn't be surprising (i.e., by heuristic explainations of how étale cohomology naturally detects such aspects of geometry).

Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ? - MathOverflow

Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ?