MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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).

Milne's Arithmetic Duality Theorems should have this.

Actually, duality theorems are better stated in the appropriate derived category and so if one is used to derived categories, then Kashiwara-Schapira (Sheaves on manifolds) should be enough to formulate the appropriate theorem.

Hope this helps.

share|cite|improve this answer
Thank you for your response. Can you give a more precise reference in Milne, Arithmetic Duality Theorems? – Timo Keller Jan 18 '12 at 18:02
Can you demonstrate it in the case of $S = \mathcal{O}_{K,S}$ ($K$ a global field) and Artin-Verdier duality? – Timo Keller Jan 18 '12 at 19:11

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.