# Separated monic representations II: Frobenius subcategories and RSS
equivalences

Research paper by **Pu Zhang, Bao-Lin Xiong**

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

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#### 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 $\x$.