Cohomology of Topologised Monoids

Abstract

We prove standard results of group cohomology -- namely, existence of a long exact sequence, classification of torsors via the first cohomology group, Shapiro's lemma, the Hochschild-Serre spectral sequence, a decomposition of the cochain complex in the direct product case, and Jannsen's result on the recovery problem -- for cohomology theories such as continuous, analytic, bounded, and pro-analytic cohomology. We also prove these results for certain monoids. The cohomology groups considered here all have very concrete interpretations by means of a cochain complex. Therefore, we do not use methods of homological algebra, but explicit calculations on the level of cochains, using techniques dating back to Hochschild and Serre.

Cited by in corpus (3)

• Finiteness of analytic cohomology of Lubin-Tate $({\phi }_L,{\Gamma }_L)$-modules
• Duality for $K$-analytic Cohomology
• Linear and quadratic Chabauty for affine hyperbolic curves