Transformation Groups

, Volume 14, Issue 3, pp 695–711 | Cite as

Finite Schur filtration dimension for modules over an algebra with Schur filtration



Let G = GLN or SLN as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2 N: Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the Hi(G, M) are Noetherian AG-modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.


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  1. [1]
    K. Akin, D. Buchsbaum, J. Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), 207–278.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    H. H. Andersen, J.-C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann. 269 (1984), 487–525.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    D. J. Benson, Representations and Cohomology. II. Cohomology of Groups and Modules, 2nd ed., Cambridge Studies in Advanced Mathematics, Vol. 31, Cambridge University Press, Cambridge, 1998.MATHGoogle Scholar
  4. [4]
    H. Borsari, W. Ferrer Santos, Geometrically reductive Hopf algebras, J. Algebra 152 (1992), 65–77.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    M. Brion, S. Kumar, Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics, Vol. 231, Birkhäuser, Boston, 2005.MATHGoogle Scholar
  6. [6]
    L. Evens, The Cohomology of Groups, Oxford Mathematical Monographs, Oxford University Press, New York, 1991.Google Scholar
  7. [7]
    E. M. Friedlander, B. J. Parshall, Cohomology of Lie algebras and algebraic groups, Amer. J. Math. 108 (1986), 235–253.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    E. M. Friedlander, A. A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), 209–270.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. A. Green, Polynomial Representations of GLn, 2nd ed., Lecture Notes in Mathematics, Vol. 830, Springer, Berlin, 2007.MATHGoogle Scholar
  10. [10]
    F. D. Grosshans, Contractions of the actions of reductive algebraic groups in arbitrary characteristic, Invent. Math. 107 (1992), 127–133.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    F. D. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory, Lecture Notes in Mathematics, Vol. 1673, Springer-Verlag, Berlin, 1997.MATHGoogle Scholar
  12. [12]
    R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russian transl.: Р. Хартсхорн, Алгебраическая геометрия, Мир, М., 1981.Google Scholar
  13. [13]
    J.-C. Jantzen, Representations of Algebraic Groups, Mathematical Surveys and Monographs, Vol. 107, Amer. Math. Soc., Providence, RI, 2003.MATHGoogle Scholar
  14. [14]
    G. Kempf, A. Ramanathan, Multicones over Schubert varieties, Invent. Math. 87 (1987), 353–363.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    W. S. Massey, Products in exact couples, Ann. of Math. 59 (1954), 558–569.CrossRefMathSciNetGoogle Scholar
  16. [16]
    M. Levine, V. Srinivas, J. Weyman, K-Theory of twisted Grassmannians, K-Theory 3 (1989), 99–121.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    O. Mathieu, Filtrations of G-modules, Ann. Sci. École Norm. Sup. 23 (1990), 625–644.MATHMathSciNetGoogle Scholar
  18. [18]
    P. Pragacz, V. Srinivas, V. Pati, Diagonal subschemes and vector bundles, available at arXiv:math.AG/0609381.
  19. [19]
    A. Touzé, Universal classes for algebraic groups, available at arXiv:0809.0989.
  20. [20]
    W. van der Kallen, Lectures on Frobenius Splittings and B-modules, Notes by S. P. Inamdar, Tata Institute of Fundamental Research, Bombay, and Springer-Verlag, Berlin, 1993.Google Scholar
  21. [21]
    W. van der Kallen, Cohomology with Grosshans graded coefficients, in: Invariant Theory in All Characteristics (H. E. A. Eddy Campbell and D. L. Wehlau eds.), CRM Proceedings and Lecture Notes, Vol. 35, Amer. Math. Soc., Providence, RI, 2004, pp. 127–138.Google Scholar
  22. [22]
    W. van der Kallen, A reductive group with finitely generated cohomology algebras, in: Algebraic Groups and Homogeneous Spaces, Mumbai 2004 (W. B. Mehta ed.), Tata Institute of Fundamental Research Studies in Mathematics, Vol. 19, Narosa, New Delhi, 2007.Google Scholar
  23. [23]
    W. van der Kallen, Finite good filtration dimension for modules over an algebra with good filtration, J. Pure Appl. Algebra 206 (2006), 59–65.MATHCrossRefMathSciNetGoogle Scholar

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© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.School of MathematicsTata Institute of Fundamental ResearchMumbaiIndia
  2. 2.Department of MathematicsUniversiteit UtrechtUtrechtThe Netherlands

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