On the Monomorphism Category of $n$-Cluster Tilting Subcategories

TL;DR

This paper explores the structure of monomorphism categories within $n$-cluster tilting subcategories of module categories over artin algebras, establishing functorial equivalences and generalizing prior work to higher-dimensional contexts.

Contribution

It constructs and analyzes functors linking submodule categories to stable categories, generalizing known results to higher $n$-cluster tilting settings and providing new applications.

Findings

Established functors are full, dense, and objective.

Derived equivalences between quotient categories of submodule and stable categories.

Connected the functors via a syzygy functor, generalizing previous results.

Abstract

Let $\mathcal{M}$ be an $n$-cluster tilting subcategory of ${\rm mod}\mbox{-}\Lambda$, where $\Lambda$ is an artin algebra. Let $\mathcal{S}(\mathcal{M})$ denotes the full subcategory of $\mathcal{S}(\Lambda)$, the submodule category of $\Lambda$, consisting of all monomorphisms in $\mathcal{M}$. We construct two functors from $\mathcal{S}(\mathcal{M})$ to ${\rm mod}\mbox{-}\underline{\mathcal{M}}$, the category of finitely presented (coherent) additive contravariant functors on the stable category of $\mathcal{M}$. We show that these functors are full, dense and objective. So they induce equivalences from the quotient categories of the submodule category of $\mathcal{M}$ modulo their respective kernels. Moreover, they are related by a syzygy functor on the stable category of ${\rm mod}\mbox{-}\underline{\mathcal{M}}$. These functors can be considered as a higher version of the two functors studied by Ringel and Zhang [RZ] in the case $\Lambda=k[x]/{\langle x^n \rangle}$ and generalized later by Eir\'{i}ksson [E] to self-injective artin algebras. Several applications will be provided.