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\section*{torsion theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{additive_and_abelian_categories}{}\paragraph*{{Additive and abelian categories}}\label{additive_and_abelian_categories}

[[!include additive and abelian categories - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{torsion_part_of_an_object}{Torsion part of an object}\dotfill \pageref*{torsion_part_of_an_object} \linebreak
\noindent\hyperlink{hereditary_torsion_theories}{Hereditary torsion theories}\dotfill \pageref*{hereditary_torsion_theories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{torsion theory} in an [[abelian category]] $A$ is a [[pair]] $(T,F)$ of [[additive category|additive]] [[subcategories]], called the \textbf{torsion class} $T$ and the \textbf{torsion free class} $F$, such that the following conditions hold:

\begin{itemize}%
\item $Hom(T,F) = 0$

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

\begin{itemize}%
\item $Hom(T,Y) = 0 \Rightarrow Y\in Ob F$


\item $Hom(X,F) = 0 \Rightarrow X\in Ob T$


\item for all $X\in Ob A$, there exists $Y\subset X$, $Y\in Ob T$ and $X/Y\in Ob F$



\end{itemize}
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.

\hypertarget{torsion_part_of_an_object}{}\subsubsection*{{Torsion part of an object}}\label{torsion_part_of_an_object}

If the [[abelian category]] $A$ satisfies the [[Pierre Gabriel|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 \emph{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.

\hypertarget{hereditary_torsion_theories}{}\subsubsection*{{Hereditary torsion theories}}\label{hereditary_torsion_theories}

A torsion theory is called \textbf{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.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

If $(T,F)$ is a torsion class then $T$ and $F$ both contain the [[zero object]] and are closed under [[biproducts]] (\href{ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra}{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]] (\href{ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra}{Borceux II 1.12.4}).

Given an [[abelian category]] $A$ there is a [[bijection]] between universal [[closure operator|closure operations]] on $A$, hereditary torsion theories in $A$ (\href{ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra}{Borceux II 1.12.8}) and, if $A$ is a [[locally finitely presentable category]] also with [[left exact functor|left exact]] [[localizations]] of $A$ admitting a [[right adjoint]] and with [[localizing subcategories]] of $A$ (\href{ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra}{Borceux II 1.13.15}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The basic example of a torsion class is the class of [[torsion subgroup|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]].

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

Comprehensive accounts are in

\begin{itemize}%
\item [[Francis Borceux]], \emph{[[Handbook of Categorical Algebra]]}, vol. 2


\item N. Popescu, \emph{Abelian categories with applications to rings and modules}, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. \href{http://www.ams.org/mathscinet-getitem?mr=0340375}{MR0340375}


\item [[Joachim Lambek]], \emph{Torsion theories, additive semantics, and rings of quotients}, with app. by H. H. Storrer on torsion theories and dominant dimensions. Lecture Notes in Mathematics \textbf{177}, Springer-Verlag 1971, vi+94 pp. \href{http://www.ams.org/mathscinet-getitem?mr=284459}{MR284459}



\end{itemize}
Historically the notion is introduced in

\begin{itemize}%
\item Spencer E. Dickson, \emph{A torsion theory for Abelian categories}, Trans. Amer. Math. Soc. \textbf{121}, No. 1 (Jan., 1966), pp. 223-235, \href{http://www.jstor.org/stable/1994341}{jstor}

\end{itemize}
For a unified treatment in Abelian and [[triangulated categories]] see

\begin{itemize}%
\item Apostolos Beligiannis, Idun Reiten, \emph{Homological and homotopical aspects of torsion theories}, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. \href{http://www.math.uoi.gr/~abeligia/torsion.pdf}{pdf}

\end{itemize}
As explained there, in triangulated context, torsion pairs are in 1-1 correspondence with [[t-structure]]s. One could also study a relation between torsion theories on an abelian category with tilting theory and $t$-structures on the derived category:

\begin{itemize}%
\item Dieter Happel, Idun Reiten, Sverre O. Smal\o{}, \emph{Tilting in abelian categories and quasitilted algebras}, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88
\item Riccardo Colpi, Luisa Fiorot, Francesco Mattiello, \emph{On tilted Giraud subcategories}, \href{http://arxiv.org/abs/1307.1987}{arxiv/1307.1987}

\end{itemize}
Other references in abelian context include

\begin{itemize}%
\item Lia Va\v{s}, \emph{Differentiability of torsion theories}, (\href{http://www.usciences.edu/~lvas/homepage_sa_umdja/difftt_single.pdf}{pdf})

\end{itemize}
For analogues in nonadditive contexts see

\begin{itemize}%
\item [[Michael Barr]], \emph{Non-abelian torsion theories}, Canad. J. Math. 25 (1973) 1224--1237
\item Basil A. Rattray, \emph{Torsion theories in non-additive categories}, Manuscripta Math. \textbf{12} (1974), 285--305 \href{http://www.ams.org/mathscinet-getitem?mr=340360}{MR340360} \href{http://dx.doi.org/10.1007/BF01155518}{doi}
\item [[Jiří Rosický ]], [[Walter Tholen]], \emph{Factorization, fibration and torsion}, \href{http://arxiv.org/abs/0801.0063}{arxiv/0801.0063}, Journal of homotopy and Related Structures
\item M. M. Clementino, D. Dikranjan, [[Walter Tholen]], \emph{Torsion theories and radicals in normal categories}, J. of Algebra \textbf{305} (2006) 92-129
\item [[Dominique Bourn]], [[Marino Gran]], \emph{Torsion theories in homological categories}, J. of Algebra \textbf{305} (2006) 18--47 \href{http://www.ams.org/mathscinet-getitem?mr=2262518}{MR2007k:18018} \href{http://dx.doi.org/10.1016/j.jalgebra.2006.07.011}{doi}

\end{itemize}
[[!redirects torsion theories]]



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