Homological Epimorphisms and the Lie Bracket in Hochschild Cohomology

Abstract

In 2009, Koenig–Nagase established a long exact sequence relating the Hochschild cohomology of an algebra with the Hochschild cohomology of the quotient of the algebra by a stratifying ideal. It is well-known that the morphisms in this long exact sequence are multiplicative. In this exposition, we will argue that those morphisms preserve the Lie bracket (and the squaring map) as well. It will turn out that this really just has to do with the fact, that the canonical map from the algebra to its quotient is a (surjective) homological epimorphism in the sense of Geigle–Lenzing. Our considerations substantially rely on a generalisation of Schwede’s homotopy theoretical interpretation of the Lie bracket in Hochschild cohomology. A brief reminder thereof will be given, too.