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, as the applications we have in mind concern $$(\varphi ,\Gamma )$$ -modules. 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.