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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.

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 |