Separated monic representations II: Frobenius subcategories and RSS equivalences

This paper aims at looking for Frobenius subcategories, via the separated monomorphism category $\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,\text{x})$, and on the other hand, to establish an {\rm RSS} equivalence from $\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,\text{x})$ to its dual $\mathrm{s}\mathrm{e}\mathrm{p}\mathrm{i}(Q,I,\text{x})$. For a bound quiver $(Q,I)$ and an algebra $A$, where $Q$ is acyclic and $I$ is generated by monomial relations, let $\mathrm{\Lambda }=A{\otimes }_{k}kQ\mathrm{/}I$. For any additive subcategory $\text{x}$ of $A$-mod, we construct $\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,\text{x})$ combinatorially. This construction describe Gorenstein-projective $\text{m}$-modules as $\mathcal {GP}(\m) = {\rm smon}(Q, I, \mathcal {GP}(A))$. It admits a homological interpretation, and enjoys a reciprocity $\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,{}^{\mathrm{\perp }}T)={}^{\mathrm{\perp }}(T\otimes kQ\mathrm{/}I)$ for a cotilting $A$-module $T$. As an application, $\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,\text{x})$ has Auslander-Reiten sequences if $\text{x}$ is resolving and contravariantly finite with $\widehat{\text{x}}=A$-mod. In particular, $\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,A)$ has Auslander-Reiten sequences. It also admits a filtration interpretation as ${\rm smon}(Q, I, \mathscr{X})={\rm Fil}(\mathscr{X}\otimes \mathcal P(kQ/I))$,providedthat$\x$isextension-closed\mathrm{.}Asanapplication,${\rm smon}(Q, I, \x)$ is an extension-closed Frobenius subcategory if and only if so is $\text{x}$. This gives “new” Frobenius subcategories of $\text{m}$-mod in the sense that they are not $\mathcal{G}\mathcal{P}(\text{m})$. 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 $\text{x}$. In particular, the Nakayama functor ${\mathcal{N}}_{\text{m}}$ gives an {\rm RSS} equivalence ${\rm smon}(Q, I, A)\cong{\rm sepi}(Q, I, A)$ifandonlyif$A$isFrobenius\mathrm{.}Forachain$Q$witharbitrary$I$, an explicit formula of an {\rm RSS} equivalence is found for arbitrary additive subcategories $\text{x}$.