The Stable Monomorphism Category of a Frobenius category

TL;DR

This paper studies the stable monomorphism category of a Frobenius category, showing it is Frobenius, relates tilting objects between categories, and establishes equivalences with singularity categories for certain algebras.

Contribution

It introduces the stable monomorphism category for Frobenius categories, demonstrates its Frobenius structure, and links it to singularity categories via tilting objects and algebraic equivalences.

Findings

The stable monomorphism category is Frobenius.

Tilting objects induce equivalences between categories.

Stable monomorphism category is equivalent to singularity category for certain algebras.

Abstract

For a Frobenius abelian category $\mathcal{A}$, we show that the category ${\rm Mon}(\mathcal{A})$ of monomorphisms in $\mathcal{A}$ is a Frobenius exact category; the associated stable category $\underline{\rm Mon}(\mathcal{A})$ modulo projective objects is called the stable monomorphism category of $\mathcal{A}$. We show that a tilting object in the stable category $\underline{\mathcal{A}}$ of $\mathcal{A}$ modulo projective objects induces naturally a tilting object in $\underline{{\rm Mon}}(\mathcal{A})$. We show that if $\mathcal{A}$ is the category of (graded) modules over a (graded) self-injective algebra $A$, then the stable monomorphism category is triangle equivalent to the (graded) singularity category of the (graded) $2\times 2$ upper triangular matrix algebra $T_2(A)$. As an application, we give two characterizations to the stable category of Ringel-Schmidmeier (\cite{RS3}).