A bimodule structure for the bounded cohomology of commutative local rings

Abstract

Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring $R$ with residue field $k$, stable cohomology modules $\widehat{\mathrm{Ext}}{\vphantom E}^{n}_R\;(k,k)$, defined for $n\in\mathbb{Z}$, have been studied by Avramov and Veliche. Stable cohomology carries a structure of $\mathbb{Z}$-graded $k$-algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology $\mathrm{Ext}_R(k,k)$ and a shift of $\mathrm{Hom}_k(\mathrm{Ext}_R(k,k),k)$. We use this information to characterize the rings $R$ for which stable cohomology is graded-commutative. Stable cohomology is connected through an exact sequence to bounded cohomology. We use this connection to understand the algebra structure of $\widehat{\mathrm{Ext}}_R(k,k)$ by investigating the structure of bounded cohomology $\overline{\mathrm{Ext}}_R(k,k)$ as a graded $\mathrm{Ext}_R(k,k)$-bimodule.