Almost-split 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).

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

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

(a1)

in , the category of finitely-generated left -modules, with the following properties:

i) and are indecomposable;

ii) the sequence does not split, i.e. there is no section of (a homomorphism such that ), or, equivalently, there is no retraction of (a homomorphism such that );

iii) given any with indecomposable and not an isomorphism, there is a lift of to (i.e. a homomorphism in such that );

iv) given any with indecomposable and not an isomorphism, there is a homomorphism such that .

Note that if iii) (or, equivalently, iv)) were to hold for all , not just those 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 , and also by . 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 be a finite-dimensional algebra over an algebraically closed field . The category is a Krull–Schmidt category (Krull–Remak–Schmidt category), i.e. a is indecomposable if and only if , the endomorphism ring of , is a local ring and (hence) the decomposition of a module in into indecomposables is unique up to isomorphism.

Let be an indecomposable and consider the contravariant functor . The morphisms that do not admit a section (i.e. an such that ) form a vector subspace . Let be the quotient functor . Then, for an indecomposable , if is isomorphic to and zero otherwise. So is a simple functor. (All functors , , are viewed as -functors, i.e. functors that take their values in the category of vector -spaces.) If is indecomposable, then (the Auslander–Reiten theorem, [a4], p.4) the simple functor admits a minimal projective resolution of the form

If is projective, is zero, otherwise is indecomposable.

If is not projective, the sequence is exact and is the almost-split sequence determined by .

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 is of finite representation type and is an almost-split sequence, then has at most terms in its decomposition into indecomposables; also, if there are indeed , then one of these is projective-injective. This can be generalized, [a7].

References

[a1]	M. Auslander, I. Reiten, "Stable equivalence of dualizing -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