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. Alternatively, we can require the last condition and the following 3: , is closed under quotients and under subobjects. It follows also that and are stable under extensions.
If the abelian category satisfies the Gabriel’s property (sup) then for every object there exist the largest subobject which is in and it is called the torsion part of (sometimes written as ). 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. For some authors (e.g. Golan) torsion theory is assumed to be hereditary.
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.
Comprehensive accounts are in
Historically the notion is introduced in
For a unified treatment in Abelian and triangulated categories see
As explained there, in triangulated context, torsion pairs are in 1-1 correspondence with t-structures. One could also study a relation between torsion theories on an abelian category with tilting theory and -structures on the derived category:
Other references in abelian context include
For analogues in nonadditive contexts see