Nekovár duality over p-adic Lie extensions of global fields

Abstract

In [Nek], Nekovař gave formulations of analogues of Tate and Poitou-Tate duality for finitely generated modules over a complete commutative local Noetherian ring R with finite residue field of characteristic a fixed prime p. In the usual formulation of these dualities, one takes the Pontryagin dual, which does not in general preserve the property of finite generation. Nekovař takes the dual with respect to a dualizing complex of Grothendieck so as to have a duality between bounded complexes of R-modules with finitely generated cohomology groups. This paper is devoted to a generalization of this result to the setting of nonabelian p-adic Lie extensions. Recall that a dualizing complex ωR is a bounded complex of R-modules with cohomology finitely generated over R that has the property that for every complex M of finitely generated R-modules, the Grothendieck dual RHomR(M,ωR) in the derived category of R-modules D(ModR) has finitely generated cohomology, and moreover, the canonical morphism M −→ RHomR(RHomR(M,ωR), ωR) is an isomorphism in D(ModR). Such a complex exists and is unique up to quasiisomorphism and translation (see [Har1]). One can choose ωR to be a bounded complex of injectives, in which case the derived homomorphism complexes are represented by the complexes of homomorphisms themselves. If R is regular, then R itself, as a complex concentrated in degree 0, is a dualizing complex, but R is not in general R-injective. If R = Zp, for instance, then the complex