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:

• $\mathrm{Hom}(T,F)=0$

(in other words $A(X,Y)=0$ if $X\in \mathrm{Ob}T$ and $Y\in \mathrm{Ob}F$.

• $\mathrm{Hom}(T,Y)=0\Rightarrow Y\in \mathrm{Ob}Y$

• $\mathrm{Hom}(X,F)=0\Rightarrow X\in \mathrm{Ob}X$

• for all $X\in \mathrm{Ob}A$, there exists $Y\subset X$, $Y\in \mathrm{Ob}T$ and $X/Y\in \mathrm{Ob}F$

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.

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 $t(X)\subset X$ called the torsion part of $X$. 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.

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 $\mathrm{Ob}A$ as an extension $0\to Y\to X\to X/Y\to 0$, $Y$ in $\mathrm{Ob}T$ by $X/Y$ in $\mathrm{Ob}F$ 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

• Francis Borceux, Handbook of categorical algebra, vol. 2
• Spencer E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121, No. 1 (Jan., 1966), pp. 223-235, jstor