Summary: Tate duality is a Pontryagin duality between the $i$th Galois cohomology group of the absolute Galois group of a local field with coefficents in a finite module and the $(2-i)$th cohomology group of the Tate twist of the Pontryagin dual of the module. Poitou-Tate duality has a similar formulation, but the duality now takes place between Galois cohomology groups of a global field with restricted ramification and compactly-supported cohomology groups. Nekovár proved analogues of these in which the module in question is a finitely generated module $T$ over a complete commutative local Noetherian ring $R$ with a commuting Galois action, or a bounded complex thereof, and the Pontryagin dual is replaced with the Grothendieck dual $T^*$, which is a bounded complex of the same form. The cochain complexes computing the Galois cohomology groups of $T$ and $T^*(1)$ are then Grothendieck dual to each other in the derived category of finitely generated $R$-modules. Given a $p$-adic Lie extension of the ground field, we extend these to dualities between Galois cochain complexes of induced modules of $T$ and $T^*(1)$ in the derived category of finitely generated modules over the possibly noncommutative Iwasawa algebra with $R$-coefficients.

11R23, 11R34, 11S25, 16E35, 18E30

Galois cohomology, Tate duality, poitou-Tate duality, Grothendieck duality