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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Classifying representations by way of Grassmannians

Author: Birge Huisgen-Zimmermann
Journal: Trans. Amer. Math. Soc. 359 (2007), 2687-2719
MSC (2000): Primary 16G10, 16G20, 16G60, 14D20, 14D22
Published electronically: January 25, 2007
MathSciNet review: 2286052
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Abstract: Let $ \Lambda$ be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $ \Lambda$ with fixed dimension $ d$ and fixed squarefree top $ T$. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of $ \Lambda$. In the case of existence of a moduli space--unexpectedly frequent in light of the stringency of fine classification--this space is always projective and, in fact, arises as a closed subvariety $ \operatorname{\mathfrak{Grass}}^T_d$ of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety $ \operatorname{\mathfrak{Grass}}^T_d$ is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of `finite local representation type at a given simple $ T$', the radical layering $ \bigl ( J^{l}M/ J^{l+1}M \bigr )_{l \ge 0}$ is shown to be a classifying invariant for the modules with top $ T$. This relies on the following general fact obtained as a byproduct: proper degenerations of a local module $ M$ never have the same radical layering as $ M$.

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Additional Information

Birge Huisgen-Zimmermann
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106

Received by editor(s): April 20, 2004
Received by editor(s) in revised form: March 21, 2005
Published electronically: January 25, 2007
Additional Notes: This research was partially supported by a grant from the National Science Foundation.
Dedicated: Dedicated to the memory of Sheila Brenner
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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