  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^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.
