Morita invariance for infinitesimal deformations (Q2149541)

It is well known that Hochschild cohomology groups of the same degree of Morita equivalent finite dimensional associative algebras are isomorphic (see [\textit{J.-L. Loday}, Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007)]). For given such algebras \(A\) and \(B\) over a field \(\mathbb K\), the authors construct an explicit ``transfer'' map \(\phi^n \colon HH^n(A) \rightarrow HH^n(B)\), \(n>0\), connecting their Hochschild cohomology groups. On the other hand, it is known that there is a bijective (one-to-one) correspondence between the space of equivalence classes of infinitesimal deformations of the first order of any associative \(\mathbb K\)-algebra and its second Hochschild cohomology group (see [\textit{M. Gerstenhaber}, Ann. Math. (2) 79, 59--103 (1964; Zbl 0123.03101)]). Let us fix a representative element \(f\) of \([f] \in HH^2(A)\) and denote by \([g]\) its image in \(HH^2(B)\) under the transfer map. The authors show that the corresponding infinitesimal deformations \(A_f\) and \(B_g\) are Morita equivalent (as associative algebras over the Study ring of dual numbers). In conclusion, they note that any finite dimensional associative algebra over an algebraically closed field is Morita equivalent to a quotient \(A\) of a path algebra and then analyze in detail this particular case. Thus, among other things, they describe the presentation by quiver and relations for the infinitesimal deformations \(A_f\) of \(A\), where \([f] \in HH^2(A)\), and discuss a series of interesting examples.