Local structure of Schelter-Procesi smooth orders

Le Bruyn, Lieven

doi:10.1090/S0002-9947-00-02567-8

Local structure of Schelter-Procesi smooth orders
HTML articles powered by AMS MathViewer

by Lieven Le Bruyn PDF

Trans. Amer. Math. Soc. 352 (2000), 4815-4841 Request permission

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

Joachim Cuntz and Daniel Quillen, Cyclic homology and nonsingularity, J. Amer. Math. Soc. 8 (1995), no. 2, 373–442. MR 1303030, DOI 10.1090/S0894-0347-1995-1303030-7
Joachim Cuntz and Daniel Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), no. 2, 251–289. MR 1303029, DOI 10.1090/S0894-0347-1995-1303029-0
Birger Iversen, Generic local structure of the morphisms in commutative algebra, Lecture Notes in Mathematics, Vol. 310, Springer-Verlag, Berlin-New York, 1973. MR 0360575, DOI 10.1007/BFb0060790
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
Lieven Le Bruyn and Claudio Procesi, Étale local structure of matrix invariants and concomitants, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 143–175. MR 911138, DOI 10.1007/BFb0079236
Lieven Le Bruyn and Claudio Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598. MR 958897, DOI 10.1090/S0002-9947-1990-0958897-0
Domingo Luna, Slices étales, Sur les groupes algébriques, Bull. Soc. Math. France, Paris, Mémoire 33, Soc. Math. France, Paris, 1973, pp. 81–105 (French). MR 0342523, DOI 10.24033/msmf.110
Claudio Procesi, A formal inverse to the Cayley-Hamilton theorem, J. Algebra 107 (1987), no. 1, 63–74. MR 883869, DOI 10.1016/0021-8693(87)90073-1
I. Reiner, Maximal orders, London Mathematical Society Monographs, No. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR 0393100
William F. Schelter, Smooth algebras, J. Algebra 103 (1986), no. 2, 677–685. MR 864437, DOI 10.1016/0021-8693(86)90160-2
Hanspeter Kraft, Peter Slodowy, and Tonny A. Springer (eds.), Algebraische Transformationsgruppen und Invariantentheorie, DMV Seminar, vol. 13, Birkhäuser Verlag, Basel, 1989 (German). MR 1044582, DOI 10.1007/978-3-0348-7662-9