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!



Relationship between topological cohomology and $\ell$-adic cohomology
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).


