A torsion theory in an abelian category is a couple of additive subcategories called the torsion class and the torsion free class such that the following conditions hold:
(in other words if and .
for all , there exists , and
Equivalently, a torsion theory in is a pair of strictly full subcategories of such that the first and last conditions in the above list hold.
If the abelian category satisfies the Gabriel’s property (sup) then for every object there exist the largest subobject called the torsion part of . Under the axiom of choice, can be extended to a functor.
A torsion theory is hereditary if is closed under subobjects, or equivalently, is left exact functor.
If is a torsion class then and both contain the zero object and are closed under biproducts (Borceux II 1.12.3). Presentation of an object in as an extension , in by in is unique up to an isomorphism of short exact sequences (Borceux II 1.12.4).
Given an abelian category there is a bijection between universal closure operations on , hereditary torsion theories in (Borceux II 1.12.8) and, if us locally finitely presentable also with left exact localizations of admiting a right adjoint and with localizing subcategories of (Borceux II 1.13.15).
The basic example of a torsion class is the class of torsion abelian groups within the category of all abelian groups. The torsion theories are often used as a means to formulate localization theory in abelian categories.