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Reductive embeddings are Cohen-Macaulay


Author: Alvaro Rittatore
Journal: Proc. Amer. Math. Soc. 131 (2003), 675-684
MSC (2000): Primary 14M17, 14M05
DOI: https://doi.org/10.1090/S0002-9939-02-06843-0
Published electronically: October 23, 2002
MathSciNet review: 1937404
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Abstract: In this paper we prove that in positive characteristics normal embeddings of connected reductive groups are Frobenius split. As a consequence, they have rational singularities and are thus Cohen-Macaulay varieties.

As an application, we study the particular case of reductive monoids, which are affine embeddings of their unit group. In particular, we show that the algebra of regular functions of a normal irreducible reductive monoid $M$ has a good filtration for the action of the unit group of $M$.


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  • [1] M. Brion and S.P. Inamdar, Frobenius splitting of spherical varieties. Proc. of Symp. in Pure Math. 56 (1994) Part I, pages 207-218. MR 95e:14037
  • [2] M. Brion and P. Polo, Large Schubert varieties. Representation Theory 4 (2000), pages 97-126. MR 2001j:14066
  • [3] S. Doty, Representation theory of reductive normal algebraic monoids. Transactions of the A.M.S. 356 N. 6 (1999), pages 2539-2551. MR 99i:20086
  • [4] C. De Concini and C. Procesi, Complete symmetric varieties. In M.F. Gherardelli, ed. Invariant Theory, Proceedings. Lect. Notes in Math. 996, pages 1-44. Springer-Verlag, 1983, New York. MR 85e:14070
  • [5] W. Fulton, Introduction to Toric Varieties, Princeton Univ. Press, 1993. MR 94g:14028
  • [6] R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag, 1977. MR 57:3116
  • [7] J.C. Jantzen, Representations of algebraic groups. Academic Press, 1987. MR 89c:20001
  • [8] F. Knop, The Luna-Vust theory of spherical embeddings. In S. Ramanan et al., editors, Proceedings of the Hydebarad Conference on Algebraic Groups, pages 225-249. National Board for Higher Mathematics, Manoj, 1991. MR 92m:14065
  • [9] N. Lauritzen, Splitting properties of complete homogeneous spaces. J. Algebra 162 (1993), no. 1, pages 178-193. MR 95f:14095
  • [10] O. Mathieu, Tilting modules and thier applications, Adv. Studies in Pure Math. 26 (2000), pages 1-68. MR 2001k:20095
  • [11] V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), pages 27-40. MR 86k:14038
  • [12] A. Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals. Publ. Math. IHES 65 (1987), pages 61-90. MR 88k:14032
  • [13] L. Renner, Cohen-Macaulay algebraic monoids. Proc. of the Amer. Math. Soc. 89, N. 4, December 1983, pages 574-578. MR 85c:20063
  • [14] A. Rittatore, Algebraic monoids and group embeddings. Trans. Groups Vol. 3, No. 4 (1998), pages 375-396. MR 2000a:14056
  • [15] E. Strickland, A vanishing theorem for group compactifications. Math. Ann. 277 (1987), pages 165-171. MR 88b:14035
  • [16] W. Van der Kallen, Frobenius splittings and $B$-modules. Springer-Verlag, 1993. MR 95i:20064
  • [17] E.B. Vinberg, On reductive algebraic semigroups. Amer. Math. Soc. Transl., Serie 2, 169 (1994), pages 145 - 182. Lie Groups and Lie Algebras. E.B. Dynkin seminar. MR 97d:20057
  • [18] T. Vust, Plongements d'espaces symétriques algébriques: une classification. Ann. Scuola Norm. Sup. Pisa, 17, 2 (1990), pages 165-194. MR 91m:14079

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Additional Information

Alvaro Rittatore
Affiliation: Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay
Email: alvaro@cmat.edu.uy

DOI: https://doi.org/10.1090/S0002-9939-02-06843-0
Keywords: Reductive embeddings, reductive monoids, Cohen-Macaulay varieties, spherical varieties
Received by editor(s): September 28, 2000
Published electronically: October 23, 2002
Additional Notes: This research was partially done during a stay at the Institut Fourier, Grenoble, France.
Communicated by: Dan M. Barbasch
Article copyright: © Copyright 2002 American Mathematical Society

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