一个自入射Koszul代数的Hochschild同调与循环同调

李兆晖, 徐运阁, 汪任

数学学报 ›› 2018, Vol. 61 ›› Issue (1) : 97-106.

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数学学报 ›› 2018, Vol. 61 ›› Issue (1) : 97-106. DOI: 10.12386/A2018sxxb0010
论文

一个自入射Koszul代数的Hochschild同调与循环同调

    李兆晖1, 徐运阁1, 汪任2
作者信息 +

Hochschild Homology and Cyclic Homology of a Self-injective Koszul Algebra

    Zhao Hui LI1, Yun Ge XU1, Ren WANG2
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文章历史 +

摘要

代数的Hochschild同调群与其对应的Gabriel箭图的循环圈有着紧密的联系.本文基于Furuya构造的一个四点自入射Koszul代数的极小投射双模分解,用组合的方法计算了该代数的Hochschild同调空间的维数,并用循环圈的语言给出该代数的Hochschild同调空间的一组k-基.进一步,当基础域k的特征为零时,我们也得到了该代数的循环同调群的维数.

Abstract

There is a close connection between Hochschild homology groups of a k-algebra and cycles of the Gabriel quiver associated to the k-algebra. In this paper,based on the minimal projective bimodule resolution of a self-injective Koszul four-point algebra constructed by Furuya, we calculate the dimensions of Hochschild homology spaces of the algebra by using combinatorial methods, and give a k-basis of every Hochschild homology space in terms of cycles. Moreover, we obtain the dimensions of cyclic homology groups of the algebra when the base field k is of zero characteristic.

关键词

Hochschild同调 / 循环圈 / 循环同调 / Koszul代数 / 自入射代数

Key words

Hochschild homology / cycle / cyclic homology / Koszul algebra / self-injective algebra

引用本文

导出引用
李兆晖, 徐运阁, 汪任. 一个自入射Koszul代数的Hochschild同调与循环同调. 数学学报, 2018, 61(1): 97-106 https://doi.org/10.12386/A2018sxxb0010
Zhao Hui LI, Yun Ge XU, Ren WANG. Hochschild Homology and Cyclic Homology of a Self-injective Koszul Algebra. Acta Mathematica Sinica, Chinese Series, 2018, 61(1): 97-106 https://doi.org/10.12386/A2018sxxb0010

参考文献

[1] Loday J. L., Cyclic Homology, Grundlehren 301, Springer, Berlin, 1992.
[2] Connes A., Noncommutative differential geometry, IHES Publ. Math., 1985, 62:257-360.
[3] Han Y., Hochschild (co)homology dimension, J. London Math. Soc., 2006, 73(2):657-668.
[4] Happel D., Hochschild cohomology of finite dimensional algebras, Lecture Notes in Math., 1989, 1404:108-126.
[5] Furuya T., Hochschild cohomology for a class of self-injective special biserial algebras of rank four, J. Pure Appl. Algebra, 2015, 219:240-254.
[6] Xu Y. G., Wang D., Hochschild (co)homology of a class of Nakayama algebras, Acta Math. Sinica, 2008, 24(7):1097-1106.
[7] Liu S. X., Zhang P., Hochschild homology of truncated algebras, Bull. London Math. Soc., 1994, 26:427-430.

基金

国家自然科学基金资助项目(11371186,11571341)

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