Frobenius Categories over a Triangular Matrix Ring and Comma Categories

TL;DR

This paper studies Frobenius categories related to triangular matrix rings and comma categories, exploring their properties and establishing recollements of stable categories with applications.

Contribution

It introduces dual notions of certain categories over triangular matrix rings, characterizes when they are Frobenius, and connects these to recollement structures and applications in comma categories.

Findings

Identifies conditions for categories to be Frobenius.

Establishes recollement of stable categories in Frobenius cases.

Provides applications to comma categories.

Abstract

We introduce the dual notions of $\mathcal{E}(\mathcal{X},M,\mathcal{Y})$ and $\mathcal{M}(\mathcal{X},M,\mathcal{Y})$, and investigate when they have enough injective objects or projective objects, when they are resolving or co-resolving, and when they are Frobenius categories. In the case that they are Frobenius categories, we establish a recollement of their stable categories. Finally some applications to comma categories are given.