# nLab torsion theory

## Definition

A torsion theory in an abelian category $A$ is a couple $(T,F)$ of additive subcategories called the torsion class $T$ and the torsion free class $F$ such that the following conditions hold:

• $Hom(T,F) = 0$

(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: $T\cap F=\emptyset$, $T$ is closed under quotients and $F$ under subobjects. It follows also that $T$ and $F$ are stable under extensions.

#### Torsion part of an object

If the abelian category 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.

#### Hereditary torsion theories

A torsion theory is 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.

## Properties

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 $A$ us locally finitely presentable also with left exact localizations of $A$ admiting a right adjoint and with localizing subcategories of $A$ (Borceux II 1.13.15).

## Examples

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.

## Literature

Comprehensive accounts are in

• Francis Borceux, Handbook of categorical algebra, vol. 2
• 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

• Spencer E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121, No. 1 (Jan., 1966), pp. 223-235, jstor

For a unified treatment in Abelian and triangulated categories see

• Apostolos Beligiannis, Idun Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. pdf

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:

• Dieter Happel, Idun Reiten, Sverre O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88
• Riccardo Colpi, Luisa Fiorot, Francesco Mattiello, On tilted Giraud subcategories, arxiv/1307.1987

Other references in abelian context include

• Lia Vaš, Differentiability of torsion theories, pdf

For analogues in nonadditive contexts see

Revised on December 16, 2014 21:02:58 by Tim Porter (127.0.0.1)