# Types of Serre subcategories of Grothendieck categories

## TL;DR

This paper classifies Serre subcategories of Grothendieck categories by assigning them unique types, explores their properties via recollements, and establishes the existence of subcategories with each type, revealing structural differences from triangulated categories.

## Contribution

It introduces a classification of Serre subcategories in Grothendieck categories by types and demonstrates the existence of subcategories for each type, using recollements and torsion theories.

## Key findings

- Each type corresponds to a unique Serre subcategory.
- Recollements of abelian categories can be extended or split under certain conditions.
- Not all properties hold in triangulated categories.

## Abstract

Every Serre subcategory of an abelian category is assigned a unique type. The type of a Serre subcategory of a Grothendieck category is in the list: $$(0, 0), \ (0, -1), \ (1, -1), \ (0, -2), \ (1, -2), \ (2, -1), \ (+\infty, -\infty);$$ and for each $(m, -n)$ in this list, there exists a Serre subcategory such that its type is $(m, -n)$. This uses right (left) recollements of abelian categories, Tachikawa-Ohtake [TO] on strongly hereditary torsion pairs, and Geigle-Lenzing [GL] on localizing subcategories. If all the functors in a recollement of abelian categories are exact, then the recollement splits. Quite surprising, any left recollement of a Grothendieck category can be extended to a recollement; but this is not true for a right recollement. Thus, a colocalizing subcategory of a Grothendieck category is localizing; but the converse is not true. All these results do not hold in triangulated categories.