Gorenstein projective objects and recollements of Abelian categories

TL;DR

This paper explores the connections of Gorenstein projective objects in three Abelian categories within a recollement, introducing n-Gorenstein tilting modules and analyzing how resolving subcategories influence global dimensions.

Contribution

It establishes how resolving subcategories relate across three Abelian categories in a recollement and introduces n-Gorenstein tilting modules in this context.

Findings

Resolving subcategories induce and are induced by other resolving subcategories.

The size relationship between the relative global dimensions of the categories.

Introduction of n-Gorenstein tilting modules and Gorenstein syzygy modules.

Abstract

In this paper, we study the relationship of Gorenstein projective objects among three Abelian categories in a recollement. As an application, we introduce the relation of $n$-Gorenstein tilting modules (and Gorenstein syzygy modules) in three Abelian categories. For a recollement of Abelian categories, we show that a resolving subcategory induce two resolving subcategories. On the other hand, we also prove that two resolving subcategories can induce a resolving subcategory. Moreover, we give the size relationship between the relative global dimensions of three Abelian categories.