Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Local structure of Schelter-Procesi smooth orders


Author: Lieven Le Bruyn
Journal: Trans. Amer. Math. Soc. 352 (2000), 4815-4841
MSC (2000): Primary 16R30
DOI: https://doi.org/10.1090/S0002-9947-00-02567-8
Published electronically: June 14, 2000
MathSciNet review: 1695028
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

In this paper we give the étale local classification of Schelter-Procesi smooth orders in central simple algebras. In particular, we prove that if $\Delta$ is a central simple $K$-algebra of dimension $n^2$, where $K$is a field of trancendence degree $d$, then there are only finitely many étale local classes of smooth orders in $\Delta$. This result is a non-commutative generalization of the fact that a smooth variety is analytically a manifold, and so has only one type of étale local behaviour.


References [Enhancements On Off] (What's this?)

  • 1. J. Cuntz and D. Quillen, Cyclic homology and nonsingularity, Journal of the AMS 8 (1995), 373-442 MR 96e:19004
  • 2. J. Cuntz and D. Quillen, Algebra extensions and nonsingularity, Journal of the AMS 8 (1995), 251-289 MR 96c:19002
  • 3. B. Iversen, Generic local structure in commutative algebra, Lecture Notes in Math. 310 Springer-Verlag, Berlin, 1973. MR 50:13023
  • 4. H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspecte der Mathematik D1 F. Vieweg, Braunschweig, 1984 MR 86j:14006
  • 5. L. Le Bruyn and C. Procesi, Etale local structure of matrix-invariants and concomitants, in Algebraic Groups, Utrecht 1986, Lecture Notes in Math. 1271, Springer-Verlag, Berlin, 1987, pp. 143-176 MR 89b:16042
  • 6. L. Le Bruyn and C. Procesi, Semi-simple representations of quivers, Trans. AMS 317 (1990) 585-598 MR 90e:16048
  • 7. D. Luna, Slices étales, Bull. Soc. Math. France, Mémoire 33 (1973) 81-105 MR 49:7269
  • 8. C. Procesi, A formal inverse to the Cayley-Hamilton theorem, J. of Algebra 107 (1987), 63-74 MR 88b:16033
  • 9. I. Reiner, Maximal orders, Academic Press (1975) MR 52:13910
  • 10. W. Schelter, Smooth algebras, J. of Algebra 103 (1986), 677-685 MR 88a:16034
  • 11. P. Slodowy, Der Scheibensatz für algebraische Transformationsgruppen, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem. 13, Birkhäuser, Basel, 1989, pp. 89-113 MR 91m:14074

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16R30

Retrieve articles in all journals with MSC (2000): 16R30


Additional Information

Lieven Le Bruyn
Affiliation: Departement Wiskunde, University of Antwerp (UIA) B.2610, Antwerp, Belgium
Email: lebruyn@wins.uia.ac.be

DOI: https://doi.org/10.1090/S0002-9947-00-02567-8
Received by editor(s): July 10, 1997
Published electronically: June 14, 2000
Additional Notes: The author is a research director of the NFWO
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society