additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
A torsion theory in an abelian category $A$ is a pair $(T,F)$ of additive subcategories, called the torsion class $T$ and the torsion free class $F$, such that the following conditions hold:
(in other words, $A(X,Y) = 0$ if $X \in Ob T$ and $Y\in Ob F$).
$Hom(T,Y) = 0 \Rightarrow Y\in Ob F$
$Hom(X,F) = 0 \Rightarrow X\in Ob T$
for all $X\in Ob A$, there exists $Y\subset X$, $Y\in Ob T$ and $X/Y\in Ob F$
Equivalently, a torsion theory in $A$ is a pair $(T,F)$ of strictly full subcategories of $A$ such that the first and last conditions in the above list hold. Alternatively, we can require the last condition and the following 3: $T\cap F=\{0\}$, $T$ is closed under quotients and $F$ under subobjects. It follows also that $T$ and $F$ are stable under extensions.
If the abelian category $A$ satisfies the Gabriel‘s property (sup) then for every object $X$ there exist the largest subobject $t(X)\subset X$ which is in $T$ and it is called the torsion part of $X$ (sometimes written as $X_T$). Under the axiom of choice, $t: X\to t(X)$ can be extended to a functor.
A torsion theory is called hereditary if $T$ is closed under subobjects, or equivalently, $t$ is left exact functor. For some authors (e.g. Golan) torsion theory is assumed to be hereditary.
If $(T,F)$ is a torsion class then $T$ and $F$ both contain the zero object and are closed under biproducts (Borceux II 1.12.3). Presentation of an object $X$ in $Ob A$ as an extension $0\to Y\to X\to X/Y\to 0$, $Y$ in $Ob T$ by $X/Y$ in $Ob F$ is unique up to an isomorphism of short exact sequences (Borceux II 1.12.4).
Given an abelian category $A$ there is a bijection between universal closure operations on $A$, hereditary torsion theories in $A$ (Borceux II 1.12.8) and, if $A$ is a locally finitely presentable category also with left exact localizations of $A$ admitting a right adjoint and with localizing subcategories of $A$ (Borceux II 1.13.15).
The basic example of a torsion class is the class of torsion abelian groups within the category $A =$ Ab of all abelian groups. The torsion theories are often used as a means to formulate localization theory in abelian categories.
Comprehensive accounts are in
N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375
Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, with app. by H. H. Storrer on torsion theories and dominant dimensions. Lecture Notes in Mathematics 177, Springer-Verlag 1971, vi+94 pp. MR284459
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 $t$-structures on the derived category:
Other references in abelian context include
For analogues in nonadditive contexts see
Last revised on May 16, 2019 at 06:32:08. See the history of this page for a list of all contributions to it.