Three results on Frobenius categories

TL;DR

This paper presents three new theoretical results on Frobenius categories, including conditions for factor categories to remain Frobenius, an embedding theorem analogue, and a method to induce Frobenius structures, with various applications.

Contribution

It introduces three novel results on Frobenius categories, expanding understanding of their structure and embeddings.

Findings

Factor categories of Frobenius categories can be Frobenius under certain conditions

Any Frobenius category can be embedded into an extension-closed subcategory of Cohen-Macaulay modules

A new Frobenius structure can be induced on categories with enough projectives and injectives

Abstract

This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen-Macaulay modules over some additive category; this is an analogue of Gabriel-Quillen's embedding theorem for Frobenius categories; (3) we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure, with which the given category becomes Frobenius. Several applications of the results are discussed.