When Ext is a Batalin–Vilkovisky algebra

  • Niels Kowalzig

    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 is Batalin–Vilkovisky if itself is a contramodule over its enveloping algebra . This is, for example, the case for symmetric algebras and Frobenius algebras with semisimple Nakayama automorphism. We also recover the construction for Hopf algebras.

Cite this article

Niels Kowalzig, When Ext is a Batalin–Vilkovisky algebra. J. Noncommut. Geom. 12 (2018), no. 3, pp. 1081–1131

DOI 10.4171/JNCG/298