# Relationship between topological cohomology and $\ell$-adic cohomology

Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral polytope, let $f$ be a Laurent polynomial in $n$-variables with coefficients in an extension of the integers and Newton polytope $\Delta$. We can view $f$ as both a Laurent polynomial over $\mathbb{C}$ and $\mathbb{F} _ q$ for some prime power $q$. Suppose that $f$ is $\Delta$-regular (for a definition, see http://www.math.uci.edu/~dwan/gottingen.pdf pg. 16) over both $\mathbb{C}$ and $\overline{\mathbb{F}} _ q$. Let $X_\mathbb{C}\subset(\mathbb{C}^*)^n$ be the affine hypersurface defined by the vanishing of $f$ in the torus (viewing $f$ as a polynomial over $\mathbb{C}$) and $X_{\overline{\mathbb{F}}_q} \subset (\overline{\mathbb{F}} _ q ^ *)^n$ be the analog over $\overline{\mathbb{F}} _ q$.

My question is:

Is it true that $$\dim (H^i(X_\mathbb{C})) = \dim H^i(X_{\overline{\mathbb{F}}_q},\mathbb{Q} _ \ell),$$ where the cohomology on the left hand side is the standard topological cohomology? If this is true, I would greatly appreciate a reference if you have one.

Thanks!

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Yes, if you avoid finitely many characteristics (away from $\ell$). Let $A$ be integers of number field, $F:X \rightarrow {\rm{Spec}}(A)$ any sep'td map of finite type. For any prime $\ell$ there is a dense open $U \subseteq {\rm{Spec}}(A)$ s.t. the constructible $\ell$-adic sheaves ${\rm{R}}^i(F_ {\ast})(\mathbf{Q}_ {\ell})$ are lisse and their formation commutes with any base change; see Deligne's "Th. finitude", SGA 4.5. Hence, for all closed pts $u \in U$, ${\rm{H}}^i(X_ {\overline{u}},\mathbf{Q}_ {\ell})$ has same dim. as for complex fibers. Now apply Artin comparison isom. to conclude. –  BCnrd Jul 31 '10 at 4:38
josh -- it would be nice if you gave more details on the notion of $\triangle$-regular. Does this imply e.g. that $X_{\bar{F}_q}$ is smooth? –  algori Jul 31 '10 at 5:00
Algori -- $\Delta$-regularity just means that if you restrict $f$ to the faces of $\Delta$ that there are no singularities of this restriction inside the torus. In particular, $f$ is a face of itself (of dimension n), so this implies that $f$ has no singular points inside the torus. Thanks Richard and BCnrd for the help! –  josh_whitney Jul 31 '10 at 5:38
josh_whitney: when you say "no singularities", do you mean over $\C$ (rather than smoothness over the ring of integers)? Can you specify away from which characteristics the zero scheme is smooth? It is tempting to then use Poincare duality to replace cohomology with compactly-supported variant (for which have an Artin comparison isom.), so can then use higher direct image sheaves with proper supports, for which base change is easier than in "Th. finitude" in SGA 4.5. (algori may have this in mind). But there remains the task of where these constr. sheaves are lisse, so feels ineffective. –  BCnrd Jul 31 '10 at 6:07

The way to study the topology of the situation was introduced by Khovanski in "Newton polyhedra, and toroidal varieties" Funkcional. Anal. i Priložen. 11 (1977), no. 4, 56--64, 96. His result (if I have interpreted it correctly) is that $X$ may be compactified as a hypersurface in a projective toric variety to a smooth variety with normal crossings such that each stratum is of the same form as $X$. As far as I can see this construction works uniformly so that we would get the same construction over a (suitable) mixed characteristic discrete valuation ring. Then desired isomorphism then follows from the smooth and proper base change theorem. (I have some vague recollection that this comment is also to be found somewhere in SGA but I am not going to do any wading looking for it...)

Addendum: Let me first note that the right setup to even formulate the question is a scheme $S$ with functions on it giving the coefficients of $f$. The latter polynomial should then be non-degenerate in the sense that all its fibres over (geometric) points of $S$ should be non-degenerate. The statement is then that if $\pi\colon X\to S$ is the scheme of zeroes of $f$ in the constant torus over $S$, then $R^i\pi_\ast\mathbb Z_\ell$ is locally constant commuting with base change where $\ell$ is invertible in $\mathcal O_S$ (together with comparison theorem of $\ell$-adic cohomology and classical for a complex point of $S$). As this statement is only dealing with the $\ell$-adic sheaves $R^i\pi_\ast\mathbb Z_{\ell}$ we may use the definition of $\ell$-adic sheaf introduced by Jouanolou in SGA V. Kohvanski's method then should give a compactification of $X$ by a smooth $S$-scheme with complement of relative normal crossings. The two theorems used, proper base change and vanishing of vanishing cycles, then follows directly from the case of finite coefficients (by Jouanolou's very definition).

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Awesome, so this would take of my concern about effectivity (and eliminate the relevance of fancier base change theorems). That is, I assume for the various strata there's a combinatorially computable invariant which determines smoothness even when the base isn't a field, and so the above geometric structure would then assure such smoothness determination identifies exactly the characteristics away from which the desired result holds. –  BCnrd Jul 31 '10 at 6:33
I thought that one of the greatest benefits of MO was that BCnrd would save every lost soul from wading through thousands of pages of EGA/SGA by instantaneously pinpointing the right section, together with all the errors and ways to correct them! –  Victor Protsak Jul 31 '10 at 8:03
Victor, as I mentioned elsewhere, as far as I know, the general Artin comparison isomorphism (for both higher direct images and higher direct images with proper supports, no smoothness hypotheses) is only proved in the published literature for torsion coefficients. The content for the $\ell$-adic case is a certain compatibility with inverse limits of sheaves entirely on the topological side (allowing constructible coefficients). My impression is that every expert works out an argument for themself, but I would be happy if someone could point to a published reference with a proof. Torsten? –  BCnrd Jul 31 '10 at 14:51
OK, my attempt at deadpan humor has spectacularly failed :( –  Victor Protsak Aug 1 '10 at 1:28

This seems to be a combination of a comparison theorem saying that for complex manifolds, ordinary topological cohomology is closely related to l-adic cohomology, and a theorem relating etale cohomology of a scheme to its reduction mod p. The standard reference for these sorts of results is the exposes by Artin in SGA IV, volume 3; all you have to do is wade through about 1000 pages of French.

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