分离幺一表示II：Frobenius子范畴与RSS等价
Separated monic representations II: Frobenius subcategories and RSS equivalences
摘要：
通过分离的单态范畴${\rm smon}(Q, I, \x)$寻找Frobenius子范畴，并建立从${\rm smon}(Q, I, \x)$到它的对偶${\rm sepi}(Q, I, \x)$的{ \rm RSS}等价.对于一个有界代数$(Q, I)$和一个代数$A$，其中$Q$是非循环的，$I$由单项式关系生成，令$Λ=A\otimes_k kQ/I$。对于$A$ -mod的任意加法子范畴$\x$，我们组合地构造了${\rm smon}(Q, I, \x)$.这种构造将Gorenstein-投射$\m$ -模描述为$\mathcal {GP}(\m) = {\rm smon}(Q, I, \mathcal {GP}(A))$。它容许同调解释，并且对于一个共倾的$A$ -模$T$，它具有一个互逆性${\rm smon}(Q, I, \ ^\bot T)= \ ^\bot (T\otimes kQ/I)$.作为一个应用，如果$\x$是可分解的，并且与$\widehat{\x}=A$ -mod逆变有限，则${\rm smon}(Q, I, \x)$有Auslander-Reiten序列，特别地，${\rm smon}(Q, I, A)$有Auslander-Reiten序列.它还允许过滤解释为${\rm smon}(Q, I, \mathscr{X})={\rm Fil}(\mathscr{X}\otimes \mathcal P(kQ/I))$，前提是$\x$是扩展封闭的。作为应用，${\rm smon}(Q, I, \x)$是扩张闭Frobenius子范畴当且仅当$\x$是扩张闭Frobenius子范畴。这给出了$\m$ -mod的“新”Frobenius子范畴，因为它们不是$\mathcal{GP}(\m)$。引入Ringel-Schmidmeier-Simson等价${\rm smon}(Q, I, \x)\cong{\rm sepi}(Q, I, \x)$，证明了任意扩张闭子范畴$\x$的存在性.特别地，中山函子$\mathcal N_\m$给出了{ \rm RSS}等价${\rm smon}(Q, I, A)\cong{\rm sepi}(Q, I, A)$当且仅当$A$是Frobenius。对于具有任意$I$的链$Q$，对于任意可加子范畴$\x$，得到了{ \rm RSS}等价的显式公式。
英文摘要：
This paper aims at looking for Frobenius subcategories, via the separated monomorphism category ${\rm smon}(Q, I, \x)$, and on the other hand, to establish an {\rm RSS} equivalence from ${\rm smon}(Q, I, \x)$ to its dual ${\rm sepi}(Q, I, \x)$. For a bound quiver $(Q, I)$ and an algebra $A$, where $Q$ is acyclic and $I$ is generated by monomial relations, let $Λ=A\otimes_k kQ/I$. For any additive subcategory $\x$ of $A$-mod, we construct ${\rm smon}(Q, I, \x)$ combinatorially. This construction describe Gorenstein-projective $\m$-modules as $\mathcal {GP}(\m) = {\rm smon}(Q, I, \mathcal {GP}(A))$. It admits a homological interpretation, and enjoys a reciprocity ${\rm smon}(Q, I, \ ^\bot T)= \ ^\bot (T\otimes kQ/I)$ for a cotilting $A$-module $T$. As an application, ${\rm smon}(Q, I, \x)$ has Auslander-Reiten sequences if $\x$ is resolving and contravariantly finite with $\widehat{\x}=A$-mod. In particular, ${\rm smon}(Q, I, A)$ has Auslander-Reiten sequences. It also admits a filtration interpretation as ${\rm smon}(Q, I, \mathscr{X})={\rm Fil}(\mathscr{X}\otimes \mathcal P(kQ/I))$, provided that $\x$ is extension-closed. As an application, ${\rm smon}(Q, I, \x)$ is an extension-closed Frobenius subcategory if and only if so is $\x$. This gives "new" Frobenius subcategories of $\m$-mod in the sense that they are not $\mathcal{GP}(\m)$. Ringel-Schmidmeier-Simson equivalence ${\rm smon}(Q, I, \x)\cong{\rm sepi}(Q, I, \x)$ is introduced and the existence is proved for arbitrary extension-closed subcategories $\x$. In particular, the Nakayama functor $\mathcal N_\m$ gives an {\rm RSS} equivalence ${\rm smon}(Q, I, A)\cong{\rm sepi}(Q, I, A)$ if and only if $A$ is Frobenius. For a chain $Q$ with arbitrary $I$, an explicit formula of an {\rm RSS} equivalence is found for arbitrary additive subcategories $\x$.
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