Frobenius morphisms over Z/p^2 and Bott vanishing

Details Frobenius morphisms over Z/p^2 and Bott vanishing Frobenius morphisms over Z/p^2 and Bott vanishing

1995-08-17

arxiv.org/abs/alg-geom/9508009v1

Mathematics

Algebraic Geometry

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Scientific paper

Let $X$ be a smooth projective algebraic variety over $Z/p$, which has a flat lift to a scheme $X'$ over $Z/p^2$. If the absolute Frobenius morphism $F$ on $X$ lifts to a morphism on $X'$, then an old trick by Mazur shows that push-down of the de Rham complex under $F$ decomposes. We show that the quasi-isomorphism in question is split. This is then applied to toric varieties (where a glueing argument gives lifting of Frobenius to $Z/p^2$) and we derive natural characteristic $p$ proofs of Bott vanishing and degeneration of the Danilov spectral sequence. For flag varieties we obtain generalizations of a result of Paranjape and Srinivas about non-lifting of Frobenius to the Witt vectors.

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