## Classifying representations by way of Grassmannians

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- by Birge Huisgen-Zimmermann PDF
- Trans. Amer. Math. Soc.
**359**(2007), 2687-2719 Request permission

## 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 $M$.## References

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## Additional Information

**Birge Huisgen-Zimmermann**- Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
- MR Author ID: 187325
- Email: birge@math.ucsb.edu
- 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.
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 2687-2719 - MSC (2000): Primary 16G10, 16G20, 16G60, 14D20, 14D22
- DOI: https://doi.org/10.1090/S0002-9947-07-03997-9
- MathSciNet review: 2286052

Dedicated: Dedicated to the memory of Sheila Brenner