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Title:
De Rham complex in positive characteristic
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Abstract:
Deligne and Illusie proved that, remarkably, de Rham cohomology of a smooth proper variety over F_p admits an analog of Hodge decomposition, provided that the variety lifts to Z/p^2 and has dimension <=p. I will discuss what can be said about de Rham cohomology of liftable varieties of arbitrary dimension. It turns out that Deligne-Illusie’s theorem continues to hold for some classes of varieties, such as F-split and quasi-F-split ones. But in general it fails — de Rham cohomology of a smooth proper liftable variety might have smaller dimension than its Hodge cohomology.
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