# combination of two duality theorems

For a variety $X$ over a finite field $k$, one combines étale Poincaré duality for $X \times_k \bar{k}$ with duality for $k$ (i.e. $H^0(k,M) \times H^1(k,M^\vee) \to H^1(k,\mathbf{Z}/n) = \mathbf{Z}/n$)

to a duality

$H^i(X,\mathcal{F}) \times H^{2d+1-i}(X,\mathcal{F}^\vee(d)) \to H^{2d+1}(X,\mathcal{F}^\vee(d)) = \mathbf{Z}/n$.

How can this be generalised, e.g. to $X/S$ with a duality on $S$?

By using a spectral sequence for the composite functor $T:\Gamma_S (f_*)$ where $f:X \to S$ is the structure morphism and $\Gamma_S$ is the global sections functor on $S$. Duality of the fibers and the base give rise to duality of the terms of the spectral sequence. The main non-formal part is in making the duality map $P: R^iT (-) \times R^jT(-)$ which involves, in particular, knowledge of the dualising object. Once you have $P$, showing that the pairing is non-degenerate etc comes from duality on the terms of the spectral sequence and the five-lemma.
The simplest instance of the non-formal part referred to above is as follows: If $$0 \to A \to B \to C\to 0$$ and $$0 \to E \to F \to G \to 0$$ are exact sequences and one has a duality pairing between $A$ and $E$ as well as between $C$ and $G$, one needs to construct a compatible map for $B$ and $F$; that the map induces a duality will follow formally in most situations (finite groups etc).
• Can you demonstrate it in the case of $S = \mathcal{O}_{K,S}$ ($K$ a global field) and Artin-Verdier duality? – user19475 Jan 18 '12 at 19:11