On the vanishing of Hochschild cohomology \(H^ 1(\Lambda,\Lambda\otimes\Lambda)\) for a local algebra \(\Lambda\) (Q2367078)

Throughout this paper we assume that \(\Lambda\) is a finite dimensional local algebra over an algebraically closed field \(K\). By considering certain subgroups of the Hochschild cohomology groups of the \(\Lambda\)-bimodule \(\Lambda \otimes \Lambda\) for a generalized biserial commutative algebra \(\Lambda\) the author proved [in J. Algebra 154, No. 2, 387-405 (1993; Zbl 0797.16010)] that \(\Lambda\) is selfinjective if and only if \(H^1(\Lambda, \Lambda \otimes \Lambda) = 0\). Here \(\Lambda\) is called generalized biserial if both composition lengths of \(_\Lambda((\text{rad }\Lambda)^i/(\text{rad }\Lambda)^{i + 1})\) and \(((\text{rad }\Lambda)^i/(\text{rad }\Lambda)^{i+1})_\Lambda \leq 2\) for all \(i = 1,2,\dots\). On the other hand for a commutative algebra \(\Lambda\) with cube zero radical using Hoshino's results \textit{H. Asashiba} proved [in Representations of finite-dimensional algebras, CMS Conf. Proc. 11, 9-23 (1991; Zbl 0753.16011)] that \(\Lambda\) is selfinjective if and only if \(\text{Ext}^1_\Lambda (_\Lambda \text{Hom}_K(\Lambda_\Lambda, K), {_\Lambda\Lambda}) \cong H^1(\Lambda, \Lambda \otimes \Lambda) = 0\). One of the purposes of this paper is to show in \S 1 that Asashiba's results together with Hoshino's can be proved directly by calculating similar subgroups of the Hochschild cohomology of the \(\Lambda\)-bimodule \(\Lambda \otimes \Lambda\) with the help of [loc. cit.].