Separated monic representations II: Frobenius subcategories and RSS equivalences

Abstract:

This paper looks for Frobenius subcategories, via the separated monomorphism category $\operatorname {smon}(Q, I, \mathscr {X})$, and on the other hand, aims to establish an RSS equivalence from $\operatorname {smon}(Q, I, \mathscr {X})$ to its dual $\operatorname {sepi}(Q, I, \mathscr {X})$. For a bound quiver $(Q, I)$ and an algebra $A$, where $Q$ is acyclic and $I$ is generated by monomial relations, let $\Lambda =A\otimes _k kQ/I$. For any additive subcategory $\mathscr {X}$ of $A$-mod, we introduce $\operatorname {smon}(Q, I, \mathscr {X})$ combinatorially. It describes Gorenstein-projective $\Lambda$-modules as $\mathcal {GP}(\Lambda ) = \operatorname {smon}(Q, I, \mathcal {GP}(A))$. It admits a homological interpretation and enjoys a reciprocity $\operatorname {smon}(Q, I, \ ^\bot T)= \ ^\bot (T\otimes kQ/I)$ for a cotilting $A$-module $T$. As an application, $\operatorname {smon}(Q, I, \mathscr {X})$ has Auslander-Reiten sequences if $\mathscr {X}$ is resolving and contravariantly finite with $\widehat {\mathscr {X}}=$𝐴-mod

. In particular, $\operatorname {smon}(Q, I, A)$ has Auslander-Reiten sequences. It also admits a filtration interpretation as $\operatorname {smon}(Q, I, \mathscr {X})=\operatorname {Fil}(\mathscr {X}\otimes \mathcal P(kQ/I))$, provided that $\mathscr {X}$ is extension-closed. As an application, $\operatorname {smon}(Q, I, \mathscr {X})$ is an extension-closed Frobenius subcategory if and only if so is $\mathscr {X}$. This gives “new” Frobenius subcategories of $\Lambda$-mod in the sense that they may not be $\mathcal {GP}(\Lambda )$. Ringel-Schmidmeier-Simson equivalence $\operatorname {smon}(Q, I, \mathscr {X})\cong \operatorname {sepi}(Q, I, \mathscr {X})$ is introduced and the existence is proved for arbitrary extension-closed subcategories $\mathscr {X}$. In particular, the Nakayama functor $\mathcal N_\Lambda$ gives an RSS equivalence $\operatorname {smon}(Q, I, A)\cong \operatorname {sepi}(Q, I, A)$ if and only if $A$ is Frobenius. For a chain $Q$ with arbitrary $I$, an explicit formula of an RSS equivalence is found for arbitrary additive subcategories $\mathscr {X}$.