Hochschild cohomology of relation extension algebras

Abstract

Let $B$ be the split extension of a finite dimensional algebra $C$ by a $C$-$C$-bimodule $E$. We define a morphism of associative graded algebras $φ^*:\HH^*(B)\rightarrow \HH^*(C)$ from the Hochschild cohomology of $B$ to that of $C$, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler. In the case of a trivial extension $B=C\ltimes E$, we give necessary and sufficient conditions for each $φ^n$ to be surjective. We prove the surjectivity of $φ^1$ for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of $φ^1$ for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.

Minor corrections. This version is close to the published version (to appear in Journal of Pure and Applied Algebra), Journal of Pure and Applied Algebra, Elsevier, 2016