单态算子与垂直算子
Monomorphism operator and perpendicular operator
摘要：
对于一个$Q$，一个$k$ -代数$A$，以及$A$ -mod的一个全子范畴$\mathcal X$，引入了单态范畴${\rm Mon}(Q, \mathcal X)$.主要结果是：如果$T$是一个$A$ -模，使得每个$T_i\in {\rm add} (T)$都有一个正合序列$0\rightarrow T_m\rightarrow...\rightarrow T_0\rightarrow D(A_A)\rightarrow 0$，则有${\rm Mon}(Q, \ ^\perp T) = \ ^\perp (kQ\otimes_k T)$ ;如果$T$是共倾的，则$kQ\otimes_k T$是一个唯一的共倾$\m$ -模，直到多重不可分解的直和数，使得${\rm Mon}(Q, \ ^\perp T)= \ ^\perp (kQ \otimes_k T)$。作为应用，若$A$是Gorenstein，则Gorenstein投射$(kQ\otimes_kA)$ -模的范畴被刻划为${\rm Mon}(Q, \mathcal{GP}(A))$，从而刻画了${\rm Mon}(Q, \mathcal X)$的逆变有限性，并给出了${\rm Mon}(Q, A)$是有限型的一个充要条件.
英文摘要：
For a quiver $Q$, a $k$-algebra $A$, and a full subcategory $\mathcal X$ of $A$-mod, the monomorphism category ${\rm Mon}(Q, \mathcal X)$ is introduced. The main result says that if $T$ is an $A$-module such that there is an exact sequence $0\rightarrow T_m\rightarrow...\rightarrow T_0\rightarrow D(A_A)\rightarrow 0$ with each $T_i\in {\rm add} (T)$, then ${\rm Mon}(Q, \ ^\perp T) = \ ^\perp (kQ\otimes_k T)$; and if $T$ is cotilting, then $kQ\otimes_k T$ is a unique cotilting $\m$-module, up to multiplicities of indecomposable direct summands, such that ${\rm Mon}(Q, \ ^\perp T)= \ ^\perp (kQ \otimes_k T)$. As applications, the category of the Gorenstein-projective $(kQ\otimes_kA)$-modules is characterized as ${\rm Mon}(Q, \mathcal{GP}(A))$ if $A$ is Gorenstein; the contravariantly finiteness of ${\rm Mon}(Q, \mathcal X)$ can be described; and a sufficient and necessary condition for ${\rm Mon}(Q, A)$ being of finite type is given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
