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A survey of existence theorems for almost split sequences. (English) Zbl 0606.16020

Representations of algebras, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 116, 81-89 (1986).
[For the entire collection see Zbl 0597.00009.]
Recall that a nonsplittable exact sequence \(0\to A\to B\to C\to 0\) in the category \({\mathcal A}\) is called an almost split sequence \((=\) Auslander- Reiten sequence) if: (I) if \(h: X\to C\) is not a splittable epimorphism then h can be lifted to B; (II) if \(t: A\to Y\) is not a splittable monomorphism then t can be extended to B. It is said that almost split sequences exist in \({\mathcal A}\) if for any indecomposable, nonprojective object C in \({\mathcal A}\) there exist in \({\mathcal A}\) an almost split sequence of the form above.
The paper contains a collection of theorems about existence of almost split sequences in various categories and about the form of the Auslander-Reiten translate in these cases. The most important categories for which the existence of almost split sequences is stated in this paper are: (1) category of finitely presented \(\Lambda\)-modules, where \(\Lambda\) is an algebra over a local, complete, noetherian ring R, finitely generated as R-module; (2) category of lattices over an R-order where R is a complete, local Gorenstein ring; and (3) category of coherent sheaves over a projective curve X, with only Gorenstein singularities. - The proofs of all results are contained in other papers of author, mainly written in collaboration with I. Reiten.
Reviewer: P.Dowbor

MSC:

16Gxx Representation theory of associative rings and algebras
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16Exx Homological methods in associative algebras
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H20 Singularities of curves, local rings

Citations:

Zbl 0597.00009