On the monomorphism category of n-cluster tilting subcategories

Abstract

Let ${\cal M}$ be an n-cluster tilting subcategory of mod-Λ, where Λ is an Artin algebra. Let ${\cal S}({\cal M})$ denote the full subcategory of ${\cal S}(\Lambda )$, the submodule category of Λ, consisting of all the monomorphisms in ${\cal M}$. We construct two functors from ${\cal S}({\cal M})$ to $\bmod - \underline {\cal M}$, the category of finitely presented additive contravariant functors on the stable category of ${\cal M}$. We show that these functors are full, dense and objective and hence provide equivalences between the quotient categories of ${\cal S}({\cal M})$ and $\bmod - \underline {\cal M}$. We also compare these two functors and show that they differ by the n-th syzygy functor, provided ${\cal M}$ is an nℤ-cluster tilting subcategory. These functors can be considered as higher versions of the two functors studied by Ringel and Zhang (2014) in the case $\Lambda = k\left[ x \right]/\left\langle {{x^n}} \right\rangle$ and generalized later by Eiríksson (2017) to self-injective Artin algebras. Several applications are provided.