[Paper Review] Separated monic representations II: Frobenius subcategories and RSS equivalences

Pu Zhang, Bao-Lin Xiong|arXiv (Cornell University)|Jul 16, 2017

Algebraic structures and combinatorial models26 references4 citations

TL;DR

This paper introduces a combinatorial construction of the separated monomorphism category ${\rm smon}(Q,I,\mathscr{X})$ for a bound quiver $(Q,I)$ with monomial relations and an algebra $A$, showing that Gorenstein-projective $Λ$-modules are precisely those in ${\rm smon}(Q,I,\mathcal{GP}(A))$. It establishes an RSS equivalence between ${\rm smon}(Q,I,\mathscr{X})$ and its dual ${\rm sepi}(Q,I,\mathscr{X})$, and proves that ${\rm smon}(Q,I,\mathscr{X})$ is a Frobenius category if and only if $\mathscr{X}$ is, yielding new Frobenius subcategories of $\u039b$-mod beyond $\mathcal{GP}(\u039b)$. The key result is a filtration interpretation ${\rm smon}(Q,I,\mathscr{X}) = {\rm Fil}(\mathscr{X} \otimes \mathcal{P}(kQ/I))$, which enables homological and duality results.

ABSTRACT

This paper aims at looking for Frobenius subcategories, via the separated monomorphism category ${ m smon}(Q, I, \x)$, and on the other hand, to establish an { m RSS} equivalence from ${ m smon}(Q, I, \x)$ to its dual ${ m sepi}(Q, I, \x)$. 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 ${ m smon}(Q, I, \x)$ combinatorially. This construction describe Gorenstein-projective $\m$-modules as $\mathcal {GP}(\m) = { m smon}(Q, I, \mathcal {GP}(A))$. It admits a homological interpretation, and enjoys a reciprocity ${ m smon}(Q, I, \ ^\bot T)= \ ^\bot (T\otimes kQ/I)$ for a cotilting $A$-module $T$. As an application, ${ m smon}(Q, I, \x)$ has Auslander-Reiten sequences if $\x$ is resolving and contravariantly finite with $\widehat{\x}=A$-mod. In particular, ${ m smon}(Q, I, A)$ has Auslander-Reiten sequences. It also admits a filtration interpretation as ${ m smon}(Q, I, \mathscr{X})={ m Fil}(\mathscr{X}\otimes \mathcal P(kQ/I))$, provided that $\x$ is extension-closed. As an application, ${ m smon}(Q, I, \x)$ 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 ${ m smon}(Q, I, \x)\cong{ m sepi}(Q, I, \x)$ is introduced and the existence is proved for arbitrary extension-closed subcategories $\x$. In particular, the Nakayama functor $\mathcal N_\m$ gives an { m RSS} equivalence ${ m smon}(Q, I, A)\cong{ m sepi}(Q, I, A)$ if and only if $A$ is Frobenius. For a chain $Q$ with arbitrary $I$, an explicit formula of an { m RSS} equivalence is found for arbitrary additive subcategories $\x$.

Motivation & Objective

• To identify extension-closed Frobenius subcategories of $\Lambda$-mod, where $\Lambda = A \otimes_k kQ/I$, by analyzing the separated monomorphism category ${\rm smon}(Q,I,\mathscr{X})$.
• To determine when ${\rm smon}(Q,I,\mathscr{X})$ is a Frobenius category, focusing on the existence of enough projective and injective objects.
• To establish a Ringel-Schmidmeier-Simson (RSS) equivalence between ${\rm smon}(Q,I,\mathscr{X})$ and its dual ${\rm sepi}(Q,I,\mathscr{X})$, generalizing duality in representation theory.
• To provide a new class of Frobenius subcategories of $\Lambda$-mod that are not equal to $\mathcal{GP}(\Lambda)$, thus extending the known structure of Gorenstein-projective modules.
• To give a homological and combinatorial interpretation of ${\rm smon}(Q,I,\mathscr{X})$ via a filtration decomposition involving projective modules over $kQ/I$.

Proposed method

• Construct ${\rm smon}(Q,I,\mathscr{X})$ combinatorially as a full subcategory of representations of the bound quiver $(Q,I)$ over $A$, where each vertex is in $\mathscr{X}$ and all arrows are monomorphisms.
• Use the filtration interpretation ${\rm smon}(Q,I,\mathscr{X}) = {\rm Fil}(\mathscr{X} \otimes \mathcal{P}(kQ/I))$ to relate the structure of ${\rm smon}(Q,I,\mathscr{X})$ to the projective modules of $kQ/I$, enabling homological control.
• Establish a reciprocity formula: ${\rm smon}(Q,I,{}^\perp T) = {}^\perp(T \otimes kQ/I)$ for a cotilting $A$-module $T$, linking cotilting theory with the monomorphism category.
• Define the RSS equivalence as a duality between ${\rm smon}(Q,I,\mathscr{X})$ and ${\rm sepi}(Q,I,\mathscr{X})$, the separated epimorphism category, using the Nakayama functor when $A$ is Frobenius.
• Prove that ${\rm smon}(Q,I,\mathscr{X})$ is an extension-closed Frobenius subcategory if and only if $\mathscr{X}$ is, via the filtration decomposition.
• Provide an explicit RSS equivalence formula for chain quivers $Q$ with arbitrary $I$, valid for any additive subcategory $\mathscr{X}$.

Experimental results

Research questions

• RQ1 When is the separated monomorphism category ${\rm smon}(Q,I,\mathscr{X})$ a Frobenius category, and what conditions on $\mathscr{X}$ ensure this?
• RQ2 How can the structure of Gorenstein-projective $\Lambda$-modules be described combinatorially via ${\rm smon}(Q,I,\mathcal{GP}(A))$?
• RQ3 Under what conditions does an RSS equivalence exist between ${\rm smon}(Q,I,\mathscr{X})$ and ${\rm sepi}(Q,I,\mathscr{X})$, and when is it induced by the Nakayama functor?
• RQ4 Can ${\rm smon}(Q,I,\mathscr{X})$ yield new Frobenius subcategories of $\Lambda$-mod that are not equal to $\mathcal{GP}(\Lambda)$?
• RQ5 What is the precise relationship between the filtration of ${\rm smon}(Q,I,\mathscr{X})$ and the tensor product $\mathscr{X} \otimes \mathcal{P}(kQ/I)$, and how does it support homological properties?

Key findings

• The Gorenstein-projective $\Lambda$-modules are exactly those in ${\rm smon}(Q,I,\mathcal{GP}(A))$, providing a combinatorial and homological description of $\mathcal{GP}(\Lambda)$.
• The category ${\rm smon}(Q,I,\mathscr{X})$ admits a filtration interpretation as ${\rm Fil}(\mathscr{X} \otimes \mathcal{P}(kQ/I))$, which is essential for proving homological properties.
• The separated monomorphism category ${\rm smon}(Q,I,\mathscr{X})$ is an extension-closed Frobenius subcategory if and only if $\mathscr{X}$ is, yielding new Frobenius subcategories of $\Lambda$-mod not equal to $\mathcal{GP}(\Lambda)$.
• An RSS equivalence exists between ${\rm smon}(Q,I,\mathscr{X})$ and ${\rm sepi}(Q,I,\mathscr{X})$ for any extension-closed subcategory $\mathscr{X}$, and is explicitly constructed.
• For a chain quiver $Q$, an explicit formula for the RSS equivalence is given, valid for arbitrary additive subcategories $\mathscr{X}$, generalizing duality in representation theory.
• The Nakayama functor induces an RSS equivalence ${\rm smon}(Q,I,A) \cong {\rm sepi}(Q,I,A)$ if and only if $A$ is Frobenius, linking duality to the Frobenius property of $A$.