Poincaré duality for smooth projective varieties over finite fields - MathOverflow

Poincaré duality for smooth projective varieties over finite fields

2010-07-08T10:26:01Z

What is exacly the statement of Poincaré duality for smooth projective varieties over finite fields and twisted constant $\mathbf{Z}_\ell$ sheaves? Where can I find a proof?

By twisted constant $\mathbf{Z}_\ell$ sheaf, I mean a system of $\mathbf{Z}/\ell^n$-sheaves that are constructible and étale locally constant, e.g. the system $(\mu_{\ell^n}) = \mathbf{Z}_\ell(1)$ .

I'm interested in the finite field case of Poincaré duality. Presumably, the formulation is something like $H^i(X, F) \times H^{2d+1-i}(X, F') \to H^{2d+1}(X, ?) = \mathbf{Z}_\ell$. Now, I want to know what $F'$ and $?$ is.

Answer by rt-ist for Poincaré duality for smooth projective varieties over finite fields

2010-07-08T12:00:01Z

I guess you know about Theorem 11.1 in Milne's book Étale cohomology. It is over a separably closed field though (i.e. not finite).

Answer by SGP for Poincaré duality for smooth projective varieties over finite fields

2011-04-19T01:35:13Z

The main case can be found in Milne's article specifically Theorems 1.13, 1.14 on page 310. The idea, briefly, is as follows: Given a sheaf $F$ on a variety $X$ over a finite field $k$, then over an algebraic closure $\bar{k}$ of $k$, the group $H^i_{et}(\bar{X}, F)$ becomes a $Gal(\bar{k}/k)$-module. There is a spectral sequence involving the $H^j(Gal(\bar{k}/k), H^i_{et}(\bar{X}, F))$ which converges to $H^n_{et}(X,F)$. This is true over any perfect field.

When you have duality over $\bar{k}$ (e.g. $X$ smooth proper and $F$ nice), combine it with duality in Galois cohomology (in our case, the group is very simple: $\hat{Z}$) to get duality over $k$. The duality theorems now reflect the $k$: if Poincare duality for $X$ of dimension $d$ over $\bar{k}$ pairs $H^i$ with $H^{2d-i}$, over $k$ the pairing will be between $H^i$ and $H^{2d +m -i}$ where $m$ is the cohomological dimension (assumed finite) of the Galois group ($m=1$ in the case of a finite field).

Hope this helps.