Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:25:39Z http://mathoverflow.net/feeds/question/22998 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22998/is-there-a-deep-relationship-between-models-and-etale-cohomology-if-so-why-a Is there a deep relationship between models and étale cohomology ? If so, why, and is it made precise somewhere ? GD 2010-04-29T16:07:15Z 2010-04-29T21:58:13Z <p>Let me recall two theorems : </p> <blockquote> <p>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 $H^i_c(X_{\overline{K}},\mathbb{Q}_{\ell})$.</p> <p>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.</p> <p>Write $K$ as the inductive limit of its $\mathbb{Z}$-sub-algebras of finite type. A <em>model</em> 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]. </p> <p><em><strong>Theorem 1</em></strong> [Serre, 2004] Under the above notations and assumptions, for a positive integer $n$, the following conditions are equivalent:</p> <ul> <li>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 $$|\mathcal{X}(s)|\equiv 0\text{ }( \text{ mod } \ell^n)$$</li> <li>For all $g\in G_K$, we have $$\text{Tr}(g,H_c^\ast(X_{\overline{K}},\mathbb{Q}_{\ell}))\equiv 0\text{ }( \text{ mod } \ell^n)$$</li> </ul> <p>This result is published in Illusie's note <em>Miscellany on traces in $\ell$-adic cohomology</em> from 2005.</p> </blockquote> <p>On the other hand, let me recall Saito's approach (1987) to stable reduction of curves : </p> <blockquote> <p>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 $p>0$. </p> <p><em><strong>Theorem 2</em></strong> [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 <em>stable model</em> 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).</p> <p>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 $Gal(\overline{K}/K)$ : </p> <p><em><strong>Theorem</em></strong> [Saito] Let $C$ be a geometrically irreducible proper smooth curve over $K$ of genus $g\geq 2$. The following conditions are equivalent : </p> <ul> <li>$C$ admits a stable model</li> <li>the action of inertia $I_K$ on $H^1(C_{\overline{K}},\mathbb{Q}_{\ell})$ is unipotent.</li> </ul> <p>I refer you to Abbes' article in <em>Courbes semi-stables et groupe fondamental en géométrie algébrique</em>, Birkhäuser (1998) for a discussion of Saito's proof.</p> </blockquote> <p><em><strong>Question</em></strong> Do you know of other theorems of a similar flavor ? Are there general remarks on why such results might not be surprising ? </p> <p>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).</p>