Poincaré duality for smooth projective varieties over finite fields 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.
Edit: One should even have for smooth separated connected varieties $U$ pure of dimension $d$ have a duality $H^i_c(U,\mathscr{F}) \times H^{2d+1-i}(U,\mathscr{F}^\vee(d)) \to H^{2d+1}_c(U,\Lambda(d)) = \Lambda$.
Is there an abstract nonsense proof using derived categories like "if there is a duality for $f$ and $g$, there is a duality for $g \circ f$" (applied to $X/\overline{\mathbf{F}_q}/\mathbf{F}_q$)?
 A: 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.  
A: 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).
A: One can adapt the proof for curves over finite fields in Milne, Étale Cohomology, Corollary V.2.3.
Is there an abstract nonsense proof using derived categories like "if there is a duality for $f$ and $g$, there is a duality for $g \circ f$" (applied to $X/\overline{\mathbf{F}_q}/\mathbf{F}_q$)?
