Galois structure of homogeneous coordinate rings
HTML articles powered by AMS MathViewer
- by Frauke M. Bleher and Ted Chinburg PDF
- Trans. Amer. Math. Soc. 360 (2008), 6269-6301
Abstract:
Suppose $G$ is a finite group acting on a projective scheme $X$ over a commutative Noetherian ring $R$. We study the $RG$-modules $\mathrm {H}^0(X,\mathcal {F} \otimes \mathcal {L}^n)$ when $n \ge 0$, and $\mathcal {F}$ and $\mathcal {L}$ are coherent $G$-sheaves on $X$ such that $\mathcal {L}$ is an ample line bundle. We show that the classes of these modules in the Grothendieck group $G_0(RG)$ of all finitely generated $RG$-modules lie in a finitely generated subgroup. Under various hypotheses, we show that there is a finite set of indecomposable $RG$-modules such that each $\mathrm {H}^0(X,\mathcal {F} \otimes \mathcal {L}^n)$ is a direct sum of these indecomposables, with multiplicities given by generalized Hilbert polynomials for $n >> 0$.References
- Gert Almkvist and Robert Fossum, Decomposition of exterior and symmetric powers of indecomposable $\textbf {Z}/p\textbf {Z}$-modules in characteristic $p$ and relations to invariants, Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), Lecture Notes in Math., vol. 641, Springer, Berlin, 1978, pp. 1–111. MR 499459
- J. Alperin and L. G. Kovacs, Periodicity of Weyl modules for $\textrm {SL}(2,\,q)$, J. Algebra 74 (1982), no. 1, 52–54. MR 644217, DOI 10.1016/0021-8693(82)90004-7
- Paul Baum, William Fulton, and George Quart, Lefschetz-Riemann-Roch for singular varieties, Acta Math. 143 (1979), no. 3-4, 193–211. MR 549774, DOI 10.1007/BF02392092
- Ted Chinburg and Boas Erez, Equivariant Euler-Poincaré characteristics and tameness, Astérisque 209 (1992), 13, 179–194. Journées Arithmétiques, 1991 (Geneva). MR 1211011
- T. Chinburg, B. Erez, G. Pappas, and M. J. Taylor, Tame actions of group schemes: integrals and slices, Duke Math. J. 82 (1996), no. 2, 269–308. MR 1387229, DOI 10.1215/S0012-7094-96-08212-5
- C. Curtis and I. Reiner, Methods of representation theory I, II. John Wiley & Sons, 1981, 1987.
- Igor Dolgachev, Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, Cambridge, 2003. MR 2004511, DOI 10.1017/CBO9780511615436
- P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
- Dan Edidin and William Graham, Riemann-Roch for equivariant Chow groups, Duke Math. J. 102 (2000), no. 3, 567–594. MR 1756110, DOI 10.1215/S0012-7094-00-10239-6
- Dan Edidin and William Graham, Nonabelian localization in equivariant $K$-theory and Riemann-Roch for quotients, Adv. Math. 198 (2005), no. 2, 547–582. MR 2183388, DOI 10.1016/j.aim.2005.06.010
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
- William Fulton and Serge Lang, Riemann-Roch algebra, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 277, Springer-Verlag, New York, 1985. MR 801033, DOI 10.1007/978-1-4757-1858-4
- D. J. Glover, A study of certain modular representations, J. Algebra 51 (1978), no. 2, 425–475. MR 476841, DOI 10.1016/0021-8693(78)90116-3
- A. Grothendieck, Étude cohomologique des faisceaux coh’erents (EGA3), Publ. Math. IHES Vols. 11 and 17, 1961 and 1963.
- A. Grothendieck, with P. Deligne, P. and N. Katz, Groupes de monodromie en géométrie algébriques (SGA 7), 1967-68. Lecture Notes in Math., Vols. 288, 340, Springer-Verlag, Heidelberg, 1972-1973.
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Dikran B. Karagueuzian and Peter Symonds, The module structure of a group action on a polynomial ring, J. Algebra 218 (1999), no. 2, 672–692. MR 1705758, DOI 10.1006/jabr.1999.7867
- D. B. Karagueuzian and P. Symonds, The module structure of a group action on a polynomial ring: A finiteness theorem. Manuscript, 2002, 20 pages.
- D. B. Karagueuzian and P. Symonds, The module structure of a group action on a polynomial ring: examples, generalizations, and applications, Invariant theory in all characteristics, CRM Proc. Lecture Notes, vol. 35, Amer. Math. Soc., Providence, RI, 2004, pp. 139–158. MR 2066462, DOI 10.1090/crmp/035/08
- Bernhard Köck, The Grothendieck-Riemann-Roch theorem for group scheme actions, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 3, 415–458 (English, with English and French summaries). MR 1621405, DOI 10.1016/S0012-9593(98)80140-7
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- S. Nakajima, On Galois module structure of the cohomology groups of an algebraic variety, Invent. Math. 75 (1984), no. 1, 1–8. MR 728135, DOI 10.1007/BF01403086
- Georgios Pappas, Galois module structure and the $\gamma$-filtration, Compositio Math. 121 (2000), no. 1, 79–104. MR 1753111, DOI 10.1023/A:1001722414377
- G. Pappas, Galois modules, ideal class groups and cubic structures, Preprint (arXiv.org/math.NT/0306309).
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
- Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
- Peter Symonds, Group action on polynomial and power series rings, Pacific J. Math. 195 (2000), no. 1, 225–230. MR 1781621, DOI 10.2140/pjm.2000.195.225
- P. Symonds, Structure theorems over polynomial rings. Manuscript, 2005, 10 pages.
- S. V. Ullom, A survey of class groups of integral group rings. In: Algebraic Number Fields (L-functions and Galois properties), 497–524, Academic Press, London-New York-San Francisco, 1977.
- Gabriele Vezzosi and Angelo Vistoli, Higher algebraic $K$-theory of group actions with finite stabilizers, Duke Math. J. 113 (2002), no. 1, 1–55. MR 1905391, DOI 10.1215/S0012-7094-02-11311-8
- Gabriele Vezzosi and Angelo Vistoli, Higher algebraic $K$-theory for actions of diagonalizable groups, Invent. Math. 153 (2003), no. 1, 1–44. MR 1990666, DOI 10.1007/s00222-002-0275-2
Additional Information
- Frauke M. Bleher
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
- Email: fbleher@math.uiowa.edu
- Ted Chinburg
- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
- Email: ted@math.upenn.edu
- Received by editor(s): May 11, 2006
- Received by editor(s) in revised form: October 26, 2006
- Published electronically: July 21, 2008
- Additional Notes: The first author was supported in part by NSF Grants DMS01-39737 and DMS06-51332 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.
- © Copyright 2008 Frauke M. Bleher and Ted Chinburg
- Journal: Trans. Amer. Math. Soc. 360 (2008), 6269-6301
- MSC (2000): Primary 14L30; Secondary 14C40, 13A50, 20C05
- DOI: https://doi.org/10.1090/S0002-9947-08-04436-X
- MathSciNet review: 2434287