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 $X/\overline{\mathbf{F}_q}/\mathbf{F}_q$)?

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. $\endgroup$curvesover finite fields. Check Milne's ADT. $\endgroup$