### Abstract

Tate duality is a Pontryagin duality between the ith 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. Nekovář 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.

Original language | English (US) |
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Pages (from-to) | 621-678 |

Number of pages | 58 |

Journal | Documenta Mathematica |

Volume | 18 |

Issue number | 2013 |

State | Published - Jan 1 2013 |

### Keywords

- Galois cohomology
- Grothendieck duality
- Poitou-Tate duality
- Tate duality

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Documenta Mathematica*,

*18*(2013), 621-678.