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 ExtˆRn(k,k), defined for n∈Z, have been studied by Avramov and Veliche. Stable cohomology carries a structure of 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 ExtR(k,k) and a shift of Homk(ExtR(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 ExtˆR(k,k) by investigating the structure of bounded cohomology Ext‾R(k,k) as a graded ExtR(k,k)-bimodule.