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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Auslander-Reiten components containing modules with bounded Betti numbers


Authors: Edward L. Green and Dan Zacharia
Journal: Trans. Amer. Math. Soc. 361 (2009), 4195-4214
MSC (2000): Primary 16G70; Secondary 16D50, 16E05
Published electronically: March 19, 2009
MathSciNet review: 2500885
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Abstract: Let $ R$ be a connected selfinjective Artin algebra, and $ M$ an indecomposable nonprojective $ R$-module with bounded Betti numbers lying in a regular component of the Auslander-Reiten quiver of $ R$. We prove that the Auslander-Reiten sequence ending at $ M$ has at most two indecomposable summands in the middle term. Furthermore we show that the component of the Auslander-Reiten quiver containing $ M$ is either a stable tube or of type $ \mathbb{Z}A_{\infty}$. We use these results to study modules with eventually constant Betti numbers, and modules with eventually periodic Betti numbers.


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Additional Information

Edward L. Green
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
Email: green@math.vt.edu

Dan Zacharia
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
Email: zacharia@syr.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04782-5
PII: S 0002-9947(09)04782-5
Received by editor(s): July 24, 2007
Published electronically: March 19, 2009
Additional Notes: Both authors are supported by grants from NSA
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.