Tensor Products of Strong Cotilting Modules and Cohen--Macaulay Algebras

报告人：黄兆泳 教授

报告题目：Tensor Products of Strong Cotilting Modules and Cohen--Macaulay Algebras

报告时间：2026年11月7日15:50-16:30

报告地点：云龙校区智华楼201报告厅

报告人简介：

黄兆泳：南京大学数学学院教授，博士生导师，主要从事同调代数和代数表示论的研究工作，曾获中国高校科学技术奖自然科学奖二等奖（两次）和江苏省数学杰出成就奖。主持国家自然科学基金面上项目多项，多次应邀在国内外重要学术会议上作大会报告。已在Israel J. Math., Publ. RIMS, Forum Math., J. Algebra, Bull. London Math. Soc., J. Pure Appl. Algebra等权威学术期刊发表论文130余篇。

报告摘要：

Let $\Lambda$ and $\Gamma$ be finite-dimensional algebras over a field $k$, and let $U$ be a $\Lambda$-module and $V$ a $\Gamma$-module.We prove that both $U$ and $V$ are strong cotilting modules if and only if $U\otimes_kV$ is a strong cotilting $\Lambda\otimes_k\Gamma$-module.As a consequence, if both $\Lambda$ and $\Gamma$ are Cohen--Macaulay algebras, then so is $\Lambda\otimes_k\Gamma$, and the converseis false in general. It answers affirmatively an open question posed by Auslander and Reiten. On the other hand, another open question posed byAuslander and Reiten is: for a Cohen--Macaulay algebra with finitistic dimension $d$, is it Iwanaga--Gorenstein if the categoryof Cohen--Macaulay modules and that of $d$-syzygy modules coincide? It was answered negatively by Chan, Iyama and Marczinzik recently,but we prove that a Cohen--Macaulay algebra with finitistic dimension $d$ is Iwanaga--Gorenstein if and only if the category ofCohen--Macaulay modules and that of $(d+1)$-syzygy modules coincide. Finally, we prove that for Cohen--Macaulay algebras$\Lambda$ and $\Gamma$, if $\Lambda\otimes_k\Gamma$ is CM-finite (respectively, CM-free), then so are both $\Lambda$ and $\Gamma$,and the converse is false in general. In particular, a conjecture posed by Hu, Luo, Xiong and Zhou is disproved.It is joint with Tiwei Zhao.