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On higher direct images of convergent isocrystals
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 11 / November 2019
- Published online by Cambridge University Press:
- 25 September 2019, pp. 2180-2213
- Print publication:
- November 2019
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Let
$k$ be a perfect field of characteristic 
$p>0$ and let 
$\operatorname{W}$ be the ring of Witt vectors of 
$k$. In this article, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over 
$k$ relative to 
$\operatorname{W}$. This proof allows us to deduce an analogue of the de Rham complexes comparison theorem of Berthelot [
$\mathscr{D}$-modules arithmétiques. II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.) 81 (2000)] without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot’s conjecture on the preservation of convergent isocrystals under the higher direct image by a smooth proper morphism of 
$k$-varieties.
