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Almost complete tilting modules. (English) Zbl 0675.16012

Let A be a basic and connected finite dimensional hereditary algebra over an algebraically closed field k. By module is meant a finitely generated module. Assume A has n non-isomorphic simple modules. Recall that a module M is called a partial tilting module if \(Ext^ 1_ A(M,M)=0\), and a tilting module if moreover it has n non-isomorphic indecomposable summands (see D. Happel and C. M. Ringel [Trans. Am. Math. Soc. 274, 399-443 (1982; Zbl 0503.16024)]). A partial tilting module M is called an almost complete tilting module if it has n-1 non-isomorphic indecomposable summands. In this latter case, an indecomposable module X is called a complement to M if \(X\not\in add M\) (where add M denotes the additive category generated by the direct summands of M), and \(X\oplus M\) is a tilting module. It is well-known that complements always exist, and it was shown by C. Riedtmann and A. Schofield [“Open orbits and their complements” (preprint)] and by L. Unger [“Schur modules over wild path algebras with three simple modules”, to appear in J. Pure Appl. Algebra] that there are at most two non-isomorphic complements. The main result of this article states that an almost complete tilting module has exactly two non-isomorphic complements if and only if it is sincere.
In order to show this result, the authors prove first that if M is an almost complete tilting module, and X, Y are two non-isomorphic complements to M, where we can assume, without loss of generality, that \(Ext^ 1_ A(Y,X)\neq 0\), then there is a non-split exact sequence \(0\to X\to E\to Y\to 0\) with \(E\in add M\). As consequences, they obtain again that almost complete tilting modules have at most two non-isomorphic indecomposable summands, and also one implication of the main theorem. After showing the other implication, they consider the case where the almost complete tilting module is not sincere, and determine the unique complement by constructing two exact sequences having this complement as middle term.
Reviewer: I.Assem

MSC:

16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras

Citations:

Zbl 0503.16024
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References:

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