---
tag: CSD6W0JV0P
kind: concept
name: Serre subcategory
msc: ["18E35", "18E10"]
specializes: [V3BFGKBJF7]
sources:
  - url: https://en.wikipedia.org/wiki/Localizing_subcategory
---

## Summary

A Serre subcategory of an [[abelian category|Y9CZ3MTNF1]] is a nonempty [[full subcategory|V3BFGKBJF7]] closed under subobjects, quotient objects and extensions.
These are the subcategories that can be "divided out".
For each Serre subcategory there is a quotient abelian category in which its objects become zero.
The objects sent to zero by an [[exact functor|S1G61TDW0K]] always form a Serre subcategory.

## Explanations

### L1 human

Think of a Serre subcategory as a class of objects that one is willing to ignore.
For this to be consistent, the class must contain every piece of its objects and everything glued together from two of its objects.
The torsion abelian groups form such a class: a subgroup or quotient of a torsion group is torsion, and so is a group built from two torsion groups.
Dividing out the torsion groups leaves a category in which a map of abelian groups counts as an isomorphism when it becomes one after tensoring with $\Q$.