ag.algebraic geometry - Poincaré duality for smooth projective varieties over finite fields

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.