DG polynomial algebras and their homological properties

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英文篇名：DG polynomial algebras and their homological properties 作者：Xuefeng ; Mao ; Xudong ; Gao ; Yanni ; Yang ; Jiahong ; Chen 英文作者：Xuefeng Mao;Xudong Gao;Yanni Yang;Jiahong Chen;Department of Mathematics, Shanghai University;Department of Mathematics, College of Mathematics and Statistics,Kashgar University; 英文关键词：DG polynomial algebra;;cohomology graded algebra;;homologically smooth;;Gorenstein;;CalabiYau 中文刊名：JAXG 英文刊名：中国科学:数学(英文版) 机构：Department of Mathematics, Shanghai University;Department of Mathematics, College of Mathematics and Statistics,Kashgar University; 出版日期：2019-03-27 出版单位：Science China(Mathematics) 年：2019 期：v.62 基金：supported by National Natural Science Foundation of China (Grant No. 11001056);; the China Postdoctoral Science Foundation (Grant Nos. 20090450066 and 201003244);; the Key Disciplines of Shanghai Municipality (Grant No. S30104);; the Innovation Program of Shanghai Municipal Education Commission (Grant No. 12YZ031) 语种：英文; 页：JAXG201904002 页数：20 CN：04 ISSN：11-5837/O1 分类号：17-36

摘要

In this paper, we introduce and study differential graded(DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A~# is a polynomial algebra K[x_1, x_2,..., x_n] with |xi| = 1 for any i ∈ {1, 2,..., n}. We describe all possible differential structures on DG polynomial algebras, compute their DG automorphism groups, study their isomorphism problems, and show that they are all homologically smooth and Gorenstein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ?_A≠ 0 and the trivial DG polynomial algebra(A, 0) is Calabi-Yau if and only if n is an odd integer.
        In this paper, we introduce and study differential graded(DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A~# is a polynomial algebra K[x_1, x_2,..., x_n] with |xi| = 1 for any i ∈ {1, 2,..., n}. We describe all possible differential structures on DG polynomial algebras, compute their DG automorphism groups, study their isomorphism problems, and show that they are all homologically smooth and Gorenstein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ?_A≠ 0 and the trivial DG polynomial algebra(A, 0) is Calabi-Yau if and only if n is an odd integer.

引文

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