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Splitting of the direct image
of sheaves under the Frobenius

Author: Rikard Bøgvad
Journal: Proc. Amer. Math. Soc. 126 (1998), 3447-3454
MSC (1991): Primary 14M25; Secondary 14F05, 14L17
MathSciNet review: 1622797
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Abstract: A generalisation and a new proof are given of a recent result of J. F. Thomsen (1996), which says that for $L$ a line bundle on a smooth toric variety $X$ over a field of positive characteristic, the direct image $F_*L$ under the Frobenius morphism splits into a direct sum of line bundles. (The special case of projective space is due to Hartshorne.) Our method is to interpret the result in terms of Grothendieck differential operators $\operatorname{Diff}^{(1)} (L,L)\cong\operatorname{Hom}_{O_{X^{(1)}}}(F_*L,F_*L)$, and $T$-linearized sheaves.

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  • [A] E. Abe, Hopf algebras, Cambridge University Press, Cambridge, 1980. MR 83a:16010
  • [B] R. Bogvad, Some results on $D$-modules, J. Algebra 173, 638-667 (1995). MR 97a:14015
  • [B1] A. Borel et al., Algebraic $D$-modules, Academic Press (1987). MR 89g:32014
  • [DG] M. Demazure and P. Gabriel, Groupes algébrique, Masson/North-Holland, Paris/Amsterdam (1970). MR 46:1800
  • [F] W. Fulton, Introduction to toric varieties, Princeton University Press (1993). MR 94g:14028
  • [HR] M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976). MR 54:5230
  • [J] J.-C. Jantzen, Representations of Algebraic Groups, Academic Press (1987). MR 89c:20001
  • [K1] T. Kaneyama, On equivariant vector bundles on an almost homogeneous variety, Nagoya Math. J. Vol. 57 (1975), 65-86. MR 51:12855
  • [K2] -, Torus-equivariant vector bundles on projective spaces, Nagoya Math. J. Vol. 111 (1988), 25-40. MR 89i:14012
  • [MR] V. B. Mehta, A. Ramanathan, Forbenious splitting and cohomology vanishing for Schubert varieties, Annals of Mathematics 122 (1985), p. 27-40. MR 86k:14038
  • [M] D. Mumford and J. Fogarty, Geometric Invariant Theory, Springer (1982). MR 86a:14006
  • [O] T. Oda, Convex Bodies and Algebraic Geometry, Berlin, Heidelberg, New York, Springer, 1988. MR 88m:14038
  • [T] J. F. Thomsen, Frobenius direct images of line bundles on toric varieties, Preprint University of Aarhus (1996).
  • [Y] A. Yekutueli, An explicit construction of the Grothendieck residue complex, Astérisque 208 (1992).

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Additional Information

Rikard Bøgvad
Affiliation: Department of Mathematics, University of Stockholm, S-106 91 Stockholm, Sweden

Received by editor(s): November 1, 1996
Communicated by: Ron Donagi
Article copyright: © Copyright 1998 American Mathematical Society

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