On the number of terms in the middle of an almost split sequence

On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

Open Mathematics ◽

10.2478/s11533-013-0328-3 ◽

2014 ◽

Vol 12 (1) ◽

Cited By ~ 2

Author(s):

Piotr Malicki ◽

José Peña ◽

Andrzej Skowroński

Keyword(s):

Module Category ◽

Artin Algebra ◽

Split Sequence ◽

Almost Split Sequence ◽

Artin Algebras ◽

Almost Split Sequences

AbstractWe prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.

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Almost Split Sequences whose Middle Term has at most Two Indecomposable Summands

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-089-5 ◽

1979 ◽

Vol 31 (5) ◽

pp. 942-960 ◽

Cited By ~ 12

Author(s):

M. Auslander ◽

R. Bautista ◽

M. I. Platzeck ◽

I. Reiten ◽

S. O. Smalø

Keyword(s):

Exact Sequence ◽

Representation Theory ◽

Finitely Generated ◽

Artin Algebra ◽

Middle Term ◽

Split Sequence ◽

Almost Split Sequence ◽

Artin Algebras ◽

Almost Split Sequences

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know thatthere exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].

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The almost split sequence starting with a simple module

Archiv der Mathematik ◽

10.1007/bf01261359 ◽

1994 ◽

Vol 62 (3) ◽

pp. 203-206 ◽

Cited By ~ 4

Author(s):

Sheila Brenner

Keyword(s):

Simple Module ◽

Split Sequence ◽

Almost Split Sequence

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Irreducible homomorphisms for lattices over orders

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270002551x ◽

1977 ◽

Vol 17 (1) ◽

pp. 109-124

Author(s):

Joachim W. Schmidt

Keyword(s):

Finite Group ◽

Direct Summand ◽

Group Ring ◽

Middle Term ◽

Split Sequence ◽

Almost Split Sequence ◽

Almost Split Sequences ◽

A Finite Group

Let Λ be a complete R-order in the semi-simple K-algebra A. Then it has been shown that for each indecomposable Λ-lattice M which is not projective, there exists a unique almost split sequence 0 → N → E → M → 0. Here we study the middle term E and characterize those almost split sequences where E has a projective direct summand. In the case where Λ is the group-ring RG for a finite group G, we get information about the almost split sequences for the syzygies and apply our results in an example.

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EXISTENCE OF ALMOST SPLIT SEQUENCES VIA REGULAR SEQUENCES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972713000099 ◽

2013 ◽

Vol 88 (2) ◽

pp. 218-231 ◽

Cited By ~ 2

Author(s):

HOSSEIN ESHRAGHI

Keyword(s):

Local Ring ◽

Endomorphism Ring ◽

Krull Dimension ◽

Inductive Argument ◽

Split Sequence ◽

Regular Sequences ◽

Complete Local Ring ◽

Almost Split Sequence ◽

Almost Split Sequences

AbstractLet $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

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