Comparing the simplicial and the Hochschild cohomologies of a finite dimensional algebra
As the title indicates, the objective of this paper is to compare the Hochschild cohomology groups \(H^i(A)=H^i(A,{_AA_A})\) of a finite dimensional basic algebra \(A\) over an algebraically closed field \(k\) with the simplicial cohomology groups \(SH^i(A,Z)\) defined for any Abelian group \(Z\) by \textit{O. Bretscher} and \textit{P. Gabriel} [in Bull. Soc. Math. Fr. 111, 21-40 (1983; Zbl 0527.16021)]. The authors define, for each \(n\), a \(k\)-linear map \(\xi_n\colon SH^n(A,k^+)\to H^n(A)\) (where \(k^+\) denotes the additive group of \(k\)), then prove that \(\xi_1\colon SH^1(A,k^+)\to H^1(A)\) is always a monomorphism, and, for \(n\geq 2\), if \(A\) is Schurian, then \(\xi_n\colon SH^n(A,k^+)\to H^n(A)\) is also a monomorphism. Applying this theorem, they show that, if \(A\) has a Galois covering \(\widetilde A\to A=\widetilde A/G\), where \(\widetilde A\) is Schurian and \(G\) is a free group, then \(\dim_kH^1(A)\geq\text{rank} G\). They also prove that, if \(A\) is a triangular Schurian algebra with global dimension at most two and \(H^2(A)=0\), then \(A\) has a multiplicative basis.
scientific article; zbMATH DE number 1308359
scientific article