Localizations of the hearts of cotorsion pairs associated with mutations

TL;DR

This paper investigates how localizations of hearts of cotorsion pairs in extriangulated categories relate under mutation, establishing a generalized pseudo-Morita equivalence between their localized categories.

Contribution

It introduces a framework for understanding the relationship between hearts of cotorsion pairs before and after mutation via a generalized pseudo-Morita equivalence.

Findings

Hearts of cotorsion pairs are equivalent to functor categories over the stable category of U.

Mutation induces a cotorsion pair (U',V') with potentially non-equivalent hearts.

A generalized pseudo-Morita equivalence links certain localizations of these hearts.

Abstract

In this article, we study localizations of hearts of cotorsion pairs (U,V) where U is rigid on an extriangulated category B. The hearts of such cotorsion pairs are equivalent to the functor categories over the stable category of U. Inspired by Marsh and Palu, we consider the mutation of U that induces a cotorsion pair (U',V'). Generally speaking, the hearts of (U,V) and (U',V') are not equivalent to each other, but we will give a generalized pseudo-Morita equivalence between certain localizations of their hearts.