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''Auslander–Reiten sequence''

Roughly speaking, almost-split sequences are minimal non-split short exact sequences. They were introduced by M. Auslander and I. Reiten in 1974–1975 and have become a central tool in the theory of representations of finite-dimensional algebras (cf. also [[Representation of an associative algebra|Representation of an associative algebra]]).

Let $R$ be an Artin algebra, ''i.e.'' $R$ is an associative ring with unity that is finitely generated as a module over its centre $Z(R)$, which is a commutative Artinian ring.

Let $C$ be an indecomposable non-projective finitely-generated left $R$-module. Then there exists a short exact sequence

\begin{equation} \tag{a1} 0 \rightarrow A \stackrel { f } { \rightarrow } B \stackrel { g } { \rightarrow } C \rightarrow 0 \end{equation}

in ${}_{R}\operatorname{Mod}$, the category of finitely-generated left $R$-modules, with the following properties:

i) $A$ and $C$ are indecomposable;

ii) the sequence does not split, ''i.e.'' there is no section $s : C \rightarrow B$ of $g$ (a homomorphism such that $g s = \operatorname{id}$), or, equivalently, there is no retraction of $f$ (a homomorphism $r : B \rightarrow A$ such that $r f = \operatorname{id}$);

iii) given any $h : Z \rightarrow C$ with $Z$ indecomposable and $h$ not an isomorphism, there is a lift of $h$ to $B$ (''i.e.'' a homomorphism $\tilde { h } : Z \rightarrow B$ in ${}_{R}\operatorname{Mod}$ such that $g \tilde { h } = h$);

iv) given any $j : A \rightarrow X$ with $X$ indecomposable and $j$ not an isomorphism, there is a homomorphism $\tilde { j } : B \rightarrow X$ such that $\tilde { j } f = j$.

Note that if iii) (or, equivalently, iv)) were to hold for all $h$, not just those $h$ that are not isomorphisms, the sequence (a1) would be split, whence  "almost split" . Moreover, a sequence (a1) with these properties is uniquely determined (up to isomorphism) by $C$, and also by $A$. This is the basic Auslander–Reiten theorem on almost-split sequences, [[#References|[a1]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]].

For convenience (things also work more generally), let now $R$ be a finite-dimensional algebra over an algebraically closed field $k$. The category $\square _ { R } \ \operatorname{Mod}$ is a Krull–Schmidt category (Krull–Remak–Schmidt category), i.e. a $C \in {}_{R}\operatorname{Mod}$ is indecomposable if and only if ${}_R\operatorname{End}(C,C)$, the endomorphism ring of $C$, is a local ring and (hence) the decomposition of a module in ${}_{R}\operatorname{Mod}$ into indecomposables is unique up to isomorphism.

Let $C$ be an indecomposable and consider the contravariant functor $X \mapsto {}_ { R } \operatorname { Mod } ( X , C )$. The morphisms $g : X \rightarrow C$ that do not admit a section (i.e. an $s : C \rightarrow X$ such that $g s = \operatorname{id}$) form a vector subspace $E _ { C } ( X ) \subset \square _ { R } \operatorname { Mod } ( X , C )$. Let $S _ { C }$ be the quotient functor $S _ { C } = \operatorname { Mod } ( ? , C ) / E _ { C }$. Then, for an indecomposable $D$, $S _ { C } ( D ) = k$ if $D$ is isomorphic to $C$ and zero otherwise. So $S _ { C }$ is a simple functor. (All functors $\Box_R \text { Mod } ( ? , C )$, $E _ { C }$, $S _ { C }$ are viewed as $k$-functors, i.e. functors that take their values in the category of vector $k$-spaces.) If $C$ is indecomposable, then (the Auslander–Reiten theorem, [[#References|[a4]]], p.4) the simple functor $S _ { C }$ admits a minimal projective resolution of the form

\begin{equation*} 0 \rightarrow {}_{R} \operatorname { Mod } ( ? , A ) \rightarrow {}_{R} \operatorname { Mod } ( ? , B ) \rightarrow \end{equation*}

\begin{equation*} \rightarrow {}_{R} \text { Mod } ( ? , C ) \rightarrow S _ { C } \rightarrow 0. \end{equation*}

If $C$ is projective, $A$ is zero, otherwise $A$ is indecomposable.

If $C$ is not projective, the sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is exact and is the almost-split sequence determined by $C$.

This functorial definition is used in [[#References|[a5]]] in the somewhat more general setting of exact categories.

For a good introduction to the use of almost-split sequences, see [[#References|[a6]]]; see also [[#References|[a3]]], [[#References|[a5]]] for comprehensive treatments. See also [[Riedtmann classification|Riedtmann classification]] for the use of almost-split sequences and the Auslander–Reiten quiver in the classification of self-injective algebras.

The Bautista–Brunner theorem says that if $R$ is of finite representation type and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is an almost-split sequence, then $B$ has at most $4$ terms in its decomposition into indecomposables; also, if there are indeed $4$, then one of these is projective-injective. This can be generalized, [[#References|[a7]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Auslander,   I. Reiten,   "Stable equivalence of dualizing $R$-varieties I"  ''Adv. Math.'' , '''12'''  (1974)  pp. 306–366</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. Auslander,   "The what, where, and why of almost split sequences" , ''Proc. ICM 1986, Berkeley'' , '''I''' , Amer. Math. Soc.  (1987)  pp. 338–345</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. Auslander,   I. Reiten,   S.O. Smalø,   "Representation theory of Artin algebras" , Cambridge Univ. Press  (1995)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  P. Gabriel,   "Auslander–Reiten sequences and representation-finite algebras"  V. Dlab (ed.)  P. Gabriel (ed.) , ''Representation Theory I. Proc. Ottawa 1979 Conf.'' , Springer  (1980)  pp. 1–71</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P. Gabriel,   A.V. Roiter,   "Representations of finite-dimensional algebras" , Springer  (1997)  pp. Sect. 9.3</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  I. Reiten,   "The use of almost split sequences in the representation theory of Artin algebras"  M. Auslander (ed.)  E. Lluis (ed.) , ''Representation of Algebras. Proc. Puebla 1978 Workshop'' , Springer  (1982)  pp. 29–104</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  Shiping Liu,   "Almost split sequenes for non-regular modules"  ''Fundam. Math.'' , '''143'''  (1993)  pp. 183–190</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  M. Auslander,   I. Reiten,   "Representation theory of Artin algebras III"  ''Commun. Algebra'' , '''3'''  (1975)  pp. 239–294</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  M. Auslander,   I. Reiten,   "Representation theory of Artin algebras IV"  ''Commun. Algebra'' , '''5'''  (1977)  pp. 443–518</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  M. Auslander,   I. Reiten,   "Representation theory of Artin algebras V"  ''Commun. Algebra'' , '''5'''  (1977)  pp. 519–554</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  M. Auslander,   I. Reiten,   "Representation theory of Artin algebras VI"  ''Commun. Algebra'' , '''6'''  (1978)  pp. 257–300</td></tr></table>