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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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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