# Journal of Noncommutative Geometry

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**Volume 12, Issue 3, 2018, pp. 1081–1131**

**DOI: 10.4171/JNCG/298**

Published online: 2018-10-30

When Ext is a Batalin–Vilkovisky algebra

Niels Kowalzig^{[1]}(1) Università di Roma La Sapienza, Rome, Italy

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.

*Keywords: *Batalin-Vilkovisky algebras, cyclic operads, Hopf algebroids, duals, Hopf-Galois maps, contramodules, Frobenius algebras, Hopf algebras, trace functors

Kowalzig Niels: When Ext is a Batalin–Vilkovisky algebra. *J. Noncommut. Geom.* 12 (2018), 1081-1131. doi: 10.4171/JNCG/298