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!

anysep'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. $\endgroup$ – BCnrd Jul 31 '10 at 4:38