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Auslander-Reiten sequences with few middle terms and applications to string algebras. (English) Zbl 0612.16013
Let us recall that if $$0\to X\to Y\to Z\to 0$$ is an Auslander-Reiten sequence of modules over an Artin algebra $$A$$ for an indecomposable nonprojective $$A$$-module $$Z$$ then the numerical invariant $$\alpha(Z)$$ is defined to be the number of indecomposable direct summands of $$Y$$. By $$\alpha(A)$$ we denote the supremum of all $$\alpha(Z)$$ for all indecomposable, nonprojective $$A$$-modules $$Z$$.
In section 1 the authors prove that for any two primitive idempotents $$e,f\in A$$ if $$0\neq a\in fJe$$ has the property that for each indecomposable direct summand $$C$$ of $$Je/Aa$$, $$C\cap\Pi(soc(Ae/Ja))\neq 0$$, where $$\Pi:Ae/Ja\to Ae/Aa$$ denotes canonical projection, [for example this holds if $$a\in fJe\setminus fJ^ 2e$$] then $$\alpha(Ae/Aa)=1$$.
In section 2 it is shown that if $${\mathcal C}$$ is a connected component of the Auslander-Reiten quiver $$\Gamma_ A$$ containing no projective and no injective modules on which $$\alpha$$ is bounded by 2, then $${\mathcal C}$$ is of the form $${\mathbb{Z}}A_{\infty}$$, $${\mathbb{Z}}A_{\infty}/<\tau^ n>$$ for some $$n\in {\mathbb{N}}_ 1$$, $${\mathbb{Z}}C_{\infty}$$ or $${\mathbb{Z}}A^{\infty}_{\infty}$$.
In section 3, using previous results, the authors give a full classification of all indecomposable modules over a string algebra $$A$$ (i.e. a biserial special algebra whose projective-injective modules are serial) and prove that then $$\alpha(A)\leq 2$$ and that almost all connected components of $$\Gamma_ A$$ are of the form $${\mathbb{Z}}A^{\infty}_{\infty}$$ and $${\mathbb{Z}}A_{\infty}/<\tau>$$.
Other proofs of the facts from the last section are contained in P. Dowbor and A. Skowroński, ”Galois coverings of representation-infinite algebras.” Comment. Math. Helv. (to appear) and in B. Wald and J. Waschbüsch, J. Algebra 95, 480-500 (1985; Zbl 0567.16007).
Reviewer: P.Dowbor

##### MSC:
 16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 16D80 Other classes of modules and ideals in associative algebras 16P10 Finite rings and finite-dimensional associative algebras 16P20 Artinian rings and modules (associative rings and algebras)
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##### References:
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