The subcategories of exact modules and RSS-equivalences
数学专题报告
报告题目(Title)：The subcategories of exact modules and RSS-equivalences
报告人(Speaker)：扶先辉（东北师范大学）
地点(Place)：后主楼1220
时间(Time)：2026年2月2日（周一）10:00-11:00
邀请人(Inviter)：胡维
报告摘要
Let $R$ be an Artin algebra, $M$ an $R$-$R$ bimodule, and $\Lambda = R\ltimes M$ the trivial extension of $R$ by $M$. The $\otimes$-exact $\Lambda$-module is defined to be the module $(X,f)$ such that $X\otimes_{R}M\otimes_{R}M\stackrel{f\otimes_{R}1_{M}}\longrightarrow X\otimes_{R}M\stackrel{f}\longrightarrow X$ is exact, whereas, the $\Hom$-exact $\Lambda$-module is defined dually, that is, it is the module $(X,f)$ such that $X\longrightarrow Hom_{R}(M,X)\longrightarrow Hom_{R}(M,Hom_{R}(M,X))$ is exact. Denoted by $\mathcal{S}(R,M)$ the subcategory of $\otimes$-exact $\Lambda$-modules, and by $\mathcal{F}(M,R)$ the subcategory of $\Hom$-exact $\Lambda$-modules. It is shown $\mathcal{S}(R,M)$ is a resolving subcategory of ${\rm mod}$-$\Lambda$ if and only if $_{R}M$ is a projective module. If $_{R}M$ and $_{R}Hom_{R}(M,R)$ are projective modules, then it is proved that $\mathcal{F}(R,M)$ is a Frobenius exact category if and only if $R$ is self-injective. Moreover, when $M\otimes_{R}M = 0$, it is shown that there is a unique cotilting right $\Lambda$-module $\mathbf{T}(DR)$, up to the multiplicity of the indecomposable direct summands, such that $\mathcal{S}(R,M) = ^{\perp}\mathbf{T}(DR)$ and, that $\mathcal{S}(M,R)$ has Auslander-Reiten sequences. Finally, if $M$ is an exchangeable bimodule, we establish the Ringel-Schmidmeier-Simson equivalence between $\mathcal{S}(M,R)$ and $\mathcal{F}(M,R)$ under the condition $M\otimes_{R}M = 0$. These generalize results given by Xiong-Zhang-Zhang and Gao-Ma-Liu. This is an ongoing work with Lifang Qin.
