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 Gabriel‘s property (sup) then for every object $X$ there exists the largest subobject $t(X)\subset X$ which is in $T$, which 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.
The notion originates with:
Spencer Ernst Dickson, Torsion theories for abelian categories, Thesis, New Mexico State University (1963) (ProQuest)
Spencer Ernst Dickson, A Torsion Theory for Abelian Categories, Transactions of the American Mathematical Society 121 1 (1966) 223-235 (doi:10.2307/1994341, jstor:1994341)
Comprehensive accounts:
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
For a unified treatment in abelian and in 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:
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
For analogues in nonadditive contexts see
Michael Barr, Non-abelian torsion theories, Canad. J. Math. 25 (1973) 1224–1237
Basil A. Rattray, Torsion theories in non-additive categories, Manuscripta Math. 12 (1974), 285–305 MR340360 doi
Jiří Rosický, Walter Tholen, Factorization, fibration and torsion, arxiv/0801.0063, Journal of homotopy and Related Structures
M. M. Clementino, D. Dikranjan, Walter Tholen, Torsion theories and radicals in normal categories, J. of Algebra 305 (2006) 92-129
Dominique Bourn, Marino Gran, Torsion theories in homological categories, J. of Algebra 305 (2006) 18–47 MR2007k:18018 doi
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