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.