User josh_whitney - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:18:06Z http://mathoverflow.net/feeds/user/8044 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33964/relationship-between-topological-cohomology-and-ell-adic-cohomology Relationship between topological cohomology and $\ell$-adic cohomology josh_whitney 2010-07-31T01:34:07Z 2010-08-01T06:49:28Z <p>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 <a href="http://www.math.uci.edu/~dwan/gottingen.pdf" rel="nofollow">http://www.math.uci.edu/~dwan/gottingen.pdf</a> 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 $\overline{\mathbb{F}} _ q$. </p> <p>My question is:</p> <p>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.</p> <p>Thanks!</p> http://mathoverflow.net/questions/33964/relationship-between-topological-cohomology-and-ell-adic-cohomology Comment by josh_whitney josh_whitney 2010-07-31T05:38:01Z 2010-07-31T05:38:01Z 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!