Alexander Petrov: On de Rham cohomology in positive characteristic
Автор: Hausdorff Center for Mathematics
Загружено: 2023-07-18
Просмотров: 502
Описание: Deligne and Illusie established an analog of Hodge decomposition in positive characteristic: for a smooth proper variety X over F_p equipped with a lift over Z/p^2, there is a natural isomorphism between de Rham and Hodge cohomology, provided that the dimension of X is at most p. It turns out that the analogous isomorphism might fail for liftable varieties of dimension larger than p (that is, the de Rham cohomology might have smaller dimension than Hodge cohomology). This failure can be seen as coming from the non-vanishing of the cohomology of reductive groups in positive characteristics combined with the different behaviour of Steenrod operations on de Rham and Hodge cohomology. I will also discuss some structures that are nonetheless always present on the de Rham complex of variety over F_p, such as the Sen operator of Drinfeld and Bhatt-Lurie in the presence of a lift over Z/p^2, and the canonical decomposition after the Frobenius pullback.
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