Almost split sequences for non-regular modules

Liu, S.

Liu, S.

Fundamenta Mathematicae, Tome 142 (1993), p. 183-190 / Harvested from The Polish Digital Mathematics Library

Résumé

Let A be an Artin algebra and let $0 \to X \to \oplus_{i = 1}^{r} Y_{i} \to Z \to 0$ be an almost split sequence of A-modules with the $Y_{i}$ indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander-Reiten quiver $Γ_{A}$ of A. Then r ≤ 4, and r = 4 implies that one of the $Y_{i}$ is projective-injective. Moreover, if $X \to \oplus_{j = 1}^{t} Y_{j}$ is a source map with the $Y_{j}$ indecomposable and X on an oriented cycle in $Γ_{A}$ , then t ≤ 4 and at most three of the $Y_{j}$ are not projective. The dual statement for a sink map holds. Finally, if an arrow X → Y in $Γ_{A}$ with valuation (d,d’) is on an oriented cycle, then dd’ ≤ 3.

Publié le : 1993-01-01

Zbl 0801.16009

EUDML-ID : urn:eudml:doc:212001

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     author = {S. Liu},
     title = {Almost split sequences for non-regular modules},
     journal = {Fundamenta Mathematicae},
     volume = {142},
     year = {1993},
     pages = {183-190},
     zbl = {0801.16009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv143i2p183bwm}
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Liu, S. Almost split sequences for non-regular modules. Fundamenta Mathematicae, Tome 142 (1993) pp. 183-190. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv143i2p183bwm/

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