Separated monic representations II: Frobenius subcategories and RSS equivalences

Research paper by Pu Zhang, Bao-Lin Xiong

Indexed on: 16 Jul '17Published on: 16 Jul '17Published in: arXiv - Mathematics - Representation Theory

Abstract

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 $\Lambda=A\otimes_k kQ/I$. 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 $\x$.