Varieties of uniserial representations IV. Kinship to geometric quotients
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- by Klaus Bongartz and Birge Huisgen-Zimmermann PDF
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Abstract:
Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field, and ${\mathbb {S}}$ a finite sequence of simple left $\Lambda$-modules. Quasiprojective subvarieties of Grassmannians, distinguished by accessible affine open covers, were introduced by the authors for use in classifying the uniserial representations of $\Lambda$ having sequence ${\mathbb {S}}$ of consecutive composition factors. Our principal objectives here are threefold: One is to prove these varieties to be ‘good approximations’—in a sense to be made precise—to geometric quotients of the (very large) classical affine varieties $\operatorname {Mod-Uni} ({\mathbb {S}})$ parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. We show that, to some extent, this fills the information gap left open by the frequent non-existence of such quotients. A second goal is that of facilitating the transfer of information among the ‘host’ varieties into which the considered quasi-projective, respectively affine, uniserial varieties are embedded. For that purpose, a general correspondence is established, between Grassmannian varieties of submodules of a projective module $P$ on one hand, and classical varieties of factor modules of $P$ on the other. Our findings are applied towards the third objective, concerning the existence of geometric quotients. The main results are then exploited in a representation-theoretic context: Among other consequences, they yield a geometric characterization of the algebras of finite uniserial type which supplements existing descriptions, but is cleaner and more readily checkable.References
- Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
- Klaus Bongartz, Gauß-Elimination und der größte gemeinsame direkte Summand von zwei endlichdimensionalen Moduln, Arch. Math. (Basel) 53 (1989), no. 3, 256–258 (German). MR 1006716, DOI 10.1007/BF01277060
- Klaus Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245–287. MR 1402728, DOI 10.1006/aima.1996.0053
- Klaus Bongartz, A note on algebras of finite uniserial type, J. Algebra 188 (1997), no. 2, 513–515. MR 1435371, DOI 10.1006/jabr.1996.6845
- Klaus Bongartz, Some geometric aspects of representation theory, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 1–27. MR 1648601
- K. Bongartz and B. Huisgen-Zimmermann, The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness, J. Pure Appl. Algebra (to appear).
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- Peter Gabriel, Finite representation type is open, Proceedings of the International Conference on Representations of Algebras (Carleton Univ., Ottawa, Ont., 1974) Carleton Math. Lecture Notes, No. 9, Carleton Univ., Ottawa, Ont., 1974, pp. 23. MR 0376769
- Birge Huisgen-Zimmermann, The geometry of uniserial representations of finite-dimensional algebra. I, J. Pure Appl. Algebra 127 (1998), no. 1, 39–72. MR 1609508, DOI 10.1016/S0022-4049(96)00184-3
- Birge Huisgen-Zimmermann, The geometry of uniserial representations of finite-dimensional algebras. III. Finite uniserial type, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4775–4812. MR 1344208, DOI 10.1090/S0002-9947-96-01575-9
- B. Huisgen-Zimmermann and A. Skowronski, The uniserial representation type of tame algebras, in preparation.
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
- A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
- Hanspeter Kraft, Geometric methods in representation theory, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 944, Springer, Berlin-New York, 1982, pp. 180–258. MR 672117
- Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 (German). MR 768181, DOI 10.1007/978-3-322-83813-1
- L. Le Bruyn, Optimal filtrations on representations of finite-dimensional algebras, Trans. Amer. Math. Soc., (to appear); see http://win-www.uia.ac.be/u/lebruyn/PAPERS/optimalnew.dvi.
- Maxwell Rosenlicht, Questions of rationality for solvable algebraic groups over nonperfect fields, Ann. Mat. Pura Appl. (4) 61 (1963), 97–120 (English, with Italian summary). MR 158891, DOI 10.1007/BF02412850
Additional Information
- Klaus Bongartz
- Affiliation: FB Mathematik, Universität Gesamthochschule Wuppertal, 42119 Wuppertal, Germany
- Email: bongartz@math.uni-wuppertal.de
- 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): December 8, 1999
- Published electronically: January 4, 2001
- Additional Notes: The research of the second author was partially supported by a National Science Foundation grant
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 2091-2113
- MSC (2000): Primary 16G10, 16G20, 16G60, 16P10
- DOI: https://doi.org/10.1090/S0002-9947-01-02712-X
- MathSciNet review: 1813609