On constructing stably equivalent functors. (English) Zbl 0810.16013
Let \(A\) be an Artin algebra, and denote by \(T(A)\) the trivial extension of \(A\) by its minimal injective cogenerator. Let \(T_ A\) be a generalized tilting module [c.f. Y. Miyashita [Math. Z. 193, 113- 146 (1986; Zbl 0578.16015)] and set \(B = \text{End}(T_ A)\). It is shown that if either \(A\) or \(B\) is representation-finite, then \(T(A)\) and \(T(B)\) are stably equivalent. This comes as a corollary of some more detailed considerations too technical to reproduce here.
Reviewer: W.H.Gustafson (Lubbock)
MSC:
16G10 | Representations of associative Artinian rings |
16D90 | Module categories in associative algebras |
16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
Keywords:
stable equivalence; Artin algebra; trivial extension; minimal injective cogenerator; generalized tilting module; representation-finiteReferences:
[1] | M. Auslander and R. O. Buchweitz; M. Auslander and R. O. Buchweitz · Zbl 0697.13005 |
[2] | Auslander, M.; Reiten, I., Stable equivalence of artin algebras, (Lecture Notes in Math, Vol. 353 (1973), Springer-Verlag: Springer-Verlag New York), 8-70 · Zbl 0276.16020 |
[3] | Auslander, M.; Reiten, I., Representation theory of artin algebras, V, Comm. Algebra, 5, 519-554 (1977) · Zbl 0396.16008 |
[4] | Auslander, M.; Smalø, S. O., Preprojective modules over artin algebras, J. Algebra, 66, 61-122 (1980) · Zbl 0477.16013 |
[5] | Happel, D.; Ringel, C. M., Tilted algebras, Trans. Amer. Math. Soc., 274, 399-443 (1982) · Zbl 0503.16024 |
[6] | Miyashita, Y., Tilting modules of finite projective dimension, Math. Z., 193, 113-146 (1986) · Zbl 0578.16015 |
[7] | Tachikawa, H.; Wakamatsu, T., Tilting functors and stable equivalences for self-injective algebras, J. Algebra, 109, 138-165 (1987) · Zbl 0616.16012 |
[8] | Wakamatsu, T., On modules with trivial self-extensions, J. Algebra, 114, 106-114 (1988) · Zbl 0646.16025 |
[9] | T. Wakamatsu; T. Wakamatsu · Zbl 0726.16009 |
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