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''Auslander–Reiten sequence''
 
''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]]).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302201.png" /> be an Artin algebra, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302202.png" /> is an associative ring with unity that is finitely generated as a module over its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302203.png" />, which is a commutative Artinian ring.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302204.png" /> be an indecomposable non-projective finitely-generated left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302205.png" />-module. Then there exists a short exact sequence
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Let $C$ be an indecomposable non-projective finitely-generated left $R$-module. Then there exists a short exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} 0 \rightarrow A \stackrel { f } { \rightarrow } B \stackrel { g } { \rightarrow } C \rightarrow 0 \end{equation}
  
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302207.png" />, the category of finitely-generated left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302208.png" />-modules, with the following properties:
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in $\square _ { R } \ \operatorname{Mod}$, the category of finitely-generated left $R$-modules, with the following properties:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a1302209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022010.png" /> are indecomposable;
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i) $A$ and $C$ are indecomposable;
  
ii) the sequence does not split, i.e. there is no section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022012.png" /> (a homomorphism such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022013.png" />), or, equivalently, there is no retraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022014.png" /> (a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022016.png" />);
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022017.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022018.png" /> indecomposable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022019.png" /> not an isomorphism, there is a lift of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022020.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022021.png" /> (i.e. a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022024.png" />);
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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 $\square _ { R } \ \operatorname{Mod}$ such that $g \tilde { h } = h$);
  
iv) given any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022026.png" /> indecomposable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022027.png" /> not an isomorphism, there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022029.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022030.png" />, not just those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022031.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022032.png" />, and also by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022033.png" />. This is the basic Auslander–Reiten theorem on almost-split sequences, [[#References|[a1]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022034.png" /> be a finite-dimensional algebra over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022035.png" />. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022036.png" /> is a Krull–Schmidt category (Krull–Remak–Schmidt category), i.e. a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022037.png" /> is indecomposable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022038.png" />, the endomorphism ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022039.png" />, is a local ring and (hence) the decomposition of a module in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022040.png" /> into indecomposables is unique up to isomorphism.
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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 $C \in \square _ { R } \operatorname{Mod}$ is indecomposable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022038.png"/>, the endomorphism ring of $C$, is a local ring and (hence) the decomposition of a module in $\square _ { R } \ \operatorname{Mod}$ into indecomposables is unique up to isomorphism.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022041.png" /> be an indecomposable and consider the contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022042.png" />. The morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022043.png" /> that do not admit a section (i.e. an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022045.png" />) form a vector subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022046.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022047.png" /> be the quotient functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022048.png" />. Then, for an indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022050.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022051.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022052.png" /> and zero otherwise. So <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022053.png" /> is a simple functor. (All functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022056.png" /> are viewed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022058.png" />-functors, i.e. functors that take their values in the category of vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022059.png" />-spaces.) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022060.png" /> is indecomposable, then (the Auslander–Reiten theorem, [[#References|[a4]]], p.4) the simple functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022061.png" /> admits a minimal projective resolution of the form
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Let $C$ be an indecomposable and consider the contravariant functor $X \mapsto \square _ { R } \operatorname { Mod } ( X , C )$. 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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022062.png" /></td> </tr></table>
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\begin{equation*} 0 \rightarrow \square _ { R } \operatorname { Mod } ( ? , A ) \rightarrow \square _ { R } \operatorname { Mod } ( ? , B ) \rightarrow \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022063.png" /></td> </tr></table>
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\begin{equation*} \rightarrow \square _ { R } \text { Mod } ( ? , C ) \rightarrow S _ { C } \rightarrow 0. \end{equation*}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022064.png" /> is projective, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022065.png" /> is zero, otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022066.png" /> is indecomposable.
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If $C$ is projective, $A$ is zero, otherwise $A$ is indecomposable.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022067.png" /> is not projective, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022068.png" /> is exact and is the almost-split sequence determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022069.png" />.
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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.
 
This functorial definition is used in [[#References|[a5]]] in the somewhat more general setting of exact categories.
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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022070.png" /> is of finite representation type and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022071.png" /> is an almost-split sequence, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022072.png" /> has at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022073.png" /> terms in its decomposition into indecomposables; also, if there are indeed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022074.png" />, then one of these is projective-injective. This can be generalized, [[#References|[a7]]].
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Auslander,  I. Reiten,  "Stable equivalence of dualizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130220/a13022075.png" />-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>
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<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>

Revision as of 16:56, 1 July 2020

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).

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 $\square _ { R } \ \operatorname{Mod}$, 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 $\square _ { R } \ \operatorname{Mod}$ 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, [a1], [a8], [a9], [a10], [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 $C \in \square _ { R } \operatorname{Mod}$ is indecomposable if and only if , the endomorphism ring of $C$, is a local ring and (hence) the decomposition of a module in $\square _ { R } \ \operatorname{Mod}$ into indecomposables is unique up to isomorphism.

Let $C$ be an indecomposable and consider the contravariant functor $X \mapsto \square _ { R } \operatorname { Mod } ( X , C )$. 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, [a4], p.4) the simple functor $S _ { C }$ admits a minimal projective resolution of the form

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

\begin{equation*} \rightarrow \square _ { 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 [a5] in the somewhat more general setting of exact categories.

For a good introduction to the use of almost-split sequences, see [a6]; see also [a3], [a5] for comprehensive treatments. See also 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, [a7].

References

[a1] M. Auslander, I. Reiten, "Stable equivalence of dualizing $R$-varieties I" Adv. Math. , 12 (1974) pp. 306–366
[a2] M. Auslander, "The what, where, and why of almost split sequences" , Proc. ICM 1986, Berkeley , I , Amer. Math. Soc. (1987) pp. 338–345
[a3] M. Auslander, I. Reiten, S.O. Smalø, "Representation theory of Artin algebras" , Cambridge Univ. Press (1995)
[a4] 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
[a5] P. Gabriel, A.V. Roiter, "Representations of finite-dimensional algebras" , Springer (1997) pp. Sect. 9.3
[a6] 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
[a7] Shiping Liu, "Almost split sequenes for non-regular modules" Fundam. Math. , 143 (1993) pp. 183–190
[a8] M. Auslander, I. Reiten, "Representation theory of Artin algebras III" Commun. Algebra , 3 (1975) pp. 239–294
[a9] M. Auslander, I. Reiten, "Representation theory of Artin algebras IV" Commun. Algebra , 5 (1977) pp. 443–518
[a10] M. Auslander, I. Reiten, "Representation theory of Artin algebras V" Commun. Algebra , 5 (1977) pp. 519–554
[a11] M. Auslander, I. Reiten, "Representation theory of Artin algebras VI" Commun. Algebra , 6 (1978) pp. 257–300
How to Cite This Entry:
Almost-split sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-split_sequence&oldid=12003
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article