Comparing the simplicial and the Hochschild cohomologies of a finite dimensional algebra
type: Item
instance of: scholarly article
title: Comparing the simplicial and the Hochschild cohomologies of a finite dimensional algebra
zbMATH Open document ID: 0940.16007
DOI: 10.1016/S0022-4049(97)00205-3
published in: Journal of Pure and Applied Algebra
publication date: 2000-07-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 1308359
zbMATH Keywords: Hochschild cohomology groups
zbMATH Keywords: finite dimensional basic algebras
zbMATH Keywords: simplicial cohomology groups
zbMATH Keywords: Galois coverings
zbMATH Keywords: Schurian algebras
zbMATH Keywords: multiplicative bases
author: Maria Izabel R. Martins
author: José A. de la Peña
reviewed by: Ibrahim Assem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308359