Morita invariance for infinitesimal deformations

Abstract

Let $A$ and $B$ be two Morita equivalent finite dimensional associative algebras over a field $\mathbb{k}$. It is well known that Hochschild cohomology is invariant under Morita equivalence. Since infinitesimal deformations are connected with the second Hochschild cohomology group, we explicitly describe the transfer map connecting ${\mathsf{HH}}^2(A)$ with ${\mathsf{HH}}^2(B)$. This allows us to transfer Morita equivalence between $A$ and $B$ to that between infinitesimal deformations of them. As an application, when $\mathbb{k}$ is algebraically closed, we consider the quotient path algebra associated to $A$ and describe the presentation by quiver and relations of the infinitesimal deformations of $A$.