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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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