Recollements induced by left Frobenius pairs

TL;DR

This paper investigates how left Frobenius pairs induce recollements in comma categories derived from abelian categories, with applications to Gorenstein homological algebra.

Contribution

It characterizes conditions under which Frobenius pairs induce recollements in comma categories, extending the theory of triangulated categories and their applications.

Findings

Recollements can be constructed from stable categories of Frobenius pairs.

Conditions for inducing Frobenius pairs in comma categories are established.

Applications include hereditary cotorsion pairs and Gorenstein projective objects.

Abstract

Given a right exact functor from an abelian category into another abelian category, there is an associated abelian category called the comma category of the functor. In this paper, we characterize when left Frobenius pairs (resp. strong left Frobenius pairs) in abelian categories can induce left Frobenius pairs (resp. strong left Frobenius pairs) in their comma categories. This leads to the construction of recollements of right triangulated categories (resp. triangulated categories) from the stable categories of left Frobenius pairs (resp. strong left Frobenius pairs). Applications are given to complete hereditary cotorsion pairs and Gorenstein projective objects.