When Ext is a Batalin–Vilkovisky algebra

Abstract

We show under what conditions the complex computing general Ext-groups carries the structure of a cyclic operad such that Ext becomes a Batalin–Vilkovisky algebra. This is achieved by transferring cyclic cohomology theories for the dual of a (left) Hopf algebroid to the complex in question, which asks for the notion of contramodules introduced along with comodules by Eilenberg–Moore half a century ago. Another crucial ingredient is an explicit formula for the inverse of the Hopf–Galois map on the dual, by which we illustrate recent categorical results and answer a long-standing open question. As an application, we prove that the Hochschild cohomology of an associative algebra $A$ is Batalin–Vilkovisky if $A$ itself is a contramodule over its enveloping algebra $A \otimes A^\mathrm {op}$. This is, for example, the case for symmetric algebras and Frobenius algebras with semisimple Nakayama automorphism. We also recover the construction for Hopf algebras.