Modules not being the middle of short chains. (Q2871302)

Let \(A\) be an Artin algebra, \(\text{mod\,}A\) the category of finitely generated right \(A\)-modules, and \(\tau_A=DTr\) the Auslander-Reiten translation on finitely generated \(A\)-modules. A sequence \(X\to M\to\tau_AX\) of non-zero homomorphisms in \(\text{mod\,}A\) with \(X\) being indecomposable is called a short chain, and \(M\) is the middle of this short chain. The class of modules which are not the middle of a short chain contains the class of directing modules, studied by several authors (see, e.g., \textit{D. Happel} and \textit{C. M. Ringel} [Arch. Math. 60, No. 3, 237-246 (1993; Zbl 0795.16007)], or \textit{A. Skowroński} [Proc. Am. Math. Soc. 120, No. 1, 19-26 (1994; Zbl 0831.16014)]).NEWLINENEWLINE In this paper, the authors give a complete description of modules in \(\text{mod\,}A\) which are not the middle of a short chain. The main result of this paper asserts that if \(A\) is an Artin algebra and \(M\) is a module in \(\text{mod\,}A\) which is not the middle of a short chain, then there exist a hereditary algebra \(H\), a tilting module \(T\) in \(\text{mod\,}H\), and an injective module \(I\) in \(\text{mod\,}H\) such that: (i) The tilted algebra \(B=\text{End}_H(T)\) is a quotient algebra of \(A\); (ii) \(M\) is isomorphic to the right \(B\)-module \(\Hom_H(T,I)\). As a consequence, it is shown that if \(A\) is an Artin algebra and \(M\) is a module in \(\text{mod\,}A\), which is not the middle term of a short chain, then \(\text{End}_A(M)\) is a hereditary algebra.