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A note on preinjective partial tilting modules. (English) Zbl 0809.16014

Dlab, Vlastimil (ed.) et al., Representations of algebras. Proceedings of the sixth international conference on representations of algebras, Carleton University, Ottawa, Ontario, Canada, August 19-22, 1992. Providence, RI: American Mathematical Society. CMS Conf. Proc. 14, 109-115 (1993).
Let \(A\) be a finite dimensional algebra over an algebraically closed field \(k\), \(A\)-mod the category of all finitely generated left \(A\)- modules, and \(D = \text{Hom}_ k (-,k)\) the standard duality on \(A\)-mod. Recall that a module \(T\) in \(A\)-mod is called a partial (generalized) tilting module if \(\text{proj.dim}_ A T < \infty\) and \(\text{Ext}^ i_ A (T,T) = 0\) for all \(i > 0\). If moreover, there exists a long exact sequence \(0 \to{_ AA}\to T_ 0 \to T_ 1 \to \dots \to T_ m \to 0\) with \(T_ i\) in \(\text{add}(T)\) for all \(i = 0, \dots, m\), then \(T\) is called a (generalized) tilting \(A\)-module. Given a partial tilting \(A\)- module \(M\), a complement to \(M\) is a module \(X\) in \(A\)-mod such that \(M \oplus X\) is a tilting \(A\)-module and \(\text{add}(M) \cap \text{add} (X)=0\). The author proves that if the length of the functor \(\text{Hom}_ A(D(A), -)\) is finite then any preinjective [in the sense of M. Auslander and S. O. Smalø, J. Algebra 66, 61-122 (1980; Zbl 0477.16013)] partial tilting \(A\)-module has a complement which is also preinjective. In the course of the proof the author gives several characterizations of algebras \(A\) such that the length of \(\text{Hom}_ A(D(A),-)\) is finite. Similar characterizations of such algebras have been also proved by the reviewer [Math. Proc. Camb. Philos. Soc. 116, 229-243 (1994)].
For the entire collection see [Zbl 0786.00022].

MSC:

16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16G10 Representations of associative Artinian rings
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16D50 Injective modules, self-injective associative rings

Citations:

Zbl 0477.16013
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