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.

provethe result one uses torsion sheaves, and hence Ext's). Or is the point of the question precisely to not invert $\ell$, and/or to encode a Galois-equivariance condition (since you mention non-sep. closed base field)? Please clarify your motivation so it is clearer what properties matter to you. – Boyarsky Jul 8 '10 at 12:59curvesover finite fields. Check Milne's ADT. – Boyarsky Jul 8 '10 at 19:13