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Glueings of tilted algebras. (English) Zbl 0821.16015

The aim of the paper is to link the theory of preprojective and preinjective partitions in the sense of M. Auslander and S. O. Smalø [J. Algebra 66, 61-122 (1980; Zbl 0477.16013)] and the tilting theory, and characterize algebras satisfying some homological conditions on modules. The authors introduce the notion of left (right) glued algebra which is a finite enlargement in the postprojective (preinjective) components of a finite set of tilted algebras having complete slices in these components. It is shown that a finite dimensional algebra \(A\) over an algebraically closed field \(k\) is left (right) glued if and only if \(\text{id}_ AM=1\) (\(\text{pd}_ A M=1\)) for all but finitely many indecomposable finite dimensional \(A\)-modules \(M\). As a consequence one gets that a representation-infinite algebra \(A\) is concealed (that is can be obtained from a hereditary algebra by a preprojective tilting module) if and only if \(\text{id}_ AM = 1\) and \(\text{pd}_ A M = 1\) for all but finitely many indecomposable finite dimensional \(A\)-modules \(M\). This characterization of concealed algebras as well as similar characterizations of algebras having at almost all indecomposable finite dimensional modules of injective (projective) dimension one have been proved also, by different methods, by the reviewer [Math. Proc. Camb. Philos. Soc. 116, 229-243 (1994; Zbl 0822.16010)]. In the paper it is also shown that, for a representation- infinite tilted algebra \(A\), \(\text{pd}_ A M = 2\) and \(\text{id}_ A M = 1\) (\(\text{pd}_ A M = 1\) and \(\text{id}_ A M = 2\)) for all but finitely many finite dimensional indecomposable \(A\)-modules \(M\) if and only if \(A\) is a left (right) glued algebra and its reduced right (left) type is a disjoint union of Dynkin graphs.

MSC:

16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
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References:

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