Hochschild (co)homology dimension

doi:10.1112/S002461070602299X

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	Hochschild (co)homology dimension
	Han, Yang
	2006-06-01
发表期刊	JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
ISSN	0024-6107
卷号	73 页码:657-668
摘要	In 1989 Happel asked the question whether, for a finite-dimensional algebra A over an algebraically closed field k, gl.dim A < infinity if and only if hch.dim A < infinity. Here, the Hochschild cohomology dimension of A is given by hch.dimA := inf{n is an element of N-0 vertical bar dimHH(i) (A) = 0 for i > n}. Recently Buchweitz, Green, Madsen and Solberg gave a negative answer to Happel's question. They found a family of pathological algebras A(q) for which gl.dim A(q) = infinity but hch.dim A(q) = 2. These algebras are pathological in many aspects. However, their Hochschild homology behaviors are not pathological any more; indeed one has hh.dimA(q) = infinity but hch.dim A(q). Here, the Hochschild homology dimension of A is given by hh.dim A := inf{n is an element of N-0 vertical bar dim HHi(A) = 0 for i > n}. This suggests posing a seemingly more reasonable conjecture by replacing the Hochschild cohomology dimension in Happel's question with the Hochschild homology dimension: gl.dim A < infinity if and only if hh.dim A < infinity if and only if hh.dim A = 0. The conjecture holds for commutative algebras and monomial algebras. In the case where A is a truncated quiver algebra, these conditions are equivalent to the condition that the quiver of A has no oriented cycles. Moreover, an algorithm for computing the Hochschild homology of any monomial algebra is provided. Thus the cyclic homology of any monomial algebra can be read off when the underlying field is characteristic 0.
DOI	10.1112/S002461070602299X
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000239040200007
出版者	LONDON MATH SOC
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/3162
专题	系统科学研究所
通讯作者	Han, Yang
作者单位	Chinese Acad Sci, Acad Math & Syst Sci, Inst Syst Sci, Beijing 100080, Peoples R China
推荐引用方式 GB/T 7714	Han, Yang. Hochschild (co)homology dimension[J]. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES,2006,73:657-668.
APA	Han, Yang.(2006).Hochschild (co)homology dimension.JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES,73,657-668.
MLA	Han, Yang."Hochschild (co)homology dimension".JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES 73(2006):657-668.