nLab
torsion theory

Definition

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

  • Hom(T,F)=0Hom(T,F) = 0

(in other words, A(X,Y)=0A(X,Y) = 0 if XObTX \in Ob T and YObFY\in Ob F).

  • Hom(T,Y)=0YObFHom(T,Y) = 0 \Rightarrow Y\in Ob F

  • Hom(X,F)=0XObTHom(X,F) = 0 \Rightarrow X\in Ob T

  • for all XObAX\in Ob A, there exists YXY\subset X, YObTY\in Ob T and X/YObFX/Y\in Ob F

Equivalently, a torsion theory in AA is a pair (T,F)(T,F) of strictly full subcategories of AA such that the first and last conditions in the above list hold. Alternatively, we can require the last condition and the following 3: TF=T\cap F=\emptyset, TT is closed under quotients and FF under subobjects. It follows also that TT and FF are stable under extensions.

Torsion part of an object

If the abelian category satisfies the Gabriel’s property (sup) then for every object XX there exist the largest subobject t(X)Xt(X)\subset X which is in TT and it is called the torsion part of XX (sometimes written as X TX_T). Under the axiom of choice, t:Xt(X)t: X\to t(X) can be extended to a functor.

Hereditary torsion theories

A torsion theory is hereditary if TT is closed under subobjects, or equivalently, tt is left exact functor. For some authors (e.g. Golan) torsion theory is assumed to be hereditary.

Properties

If (T,F)(T,F) is a torsion class then TT and FF both contain the zero object and are closed under biproducts (Borceux II 1.12.3). Presentation of an object XX in ObAOb A as an extension 0YXX/Y00\to Y\to X\to X/Y\to 0, YY in ObTOb T by X/YX/Y in ObFOb F is unique up to an isomorphism of short exact sequences (Borceux II 1.12.4).

Given an abelian category AA there is a bijection between universal closure operations on AA, hereditary torsion theories in AA (Borceux II 1.12.8) and, if AA us locally finitely presentable also with left exact localizations of AA admiting a right adjoint and with localizing subcategories of AA (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 tt-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)