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Frobenius splitting and ordinarity. (English) Zbl 1074.14019

Summary: We examine the relationship between the notion of Frobenius splitting and ordinarity for varieties. We show that the de Rham-Witt cohomology groups \(H^{i} (X,W({\mathcal O}_{X}))\) of a smooth projective Frobenius split variety are finitely generated over \(W(k)\). We provide counterexamples to a conjecture of Mehta that Frobenius split varieties are ordinary or even Hodge-Witt.

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14G27 Other nonalgebraically closed ground fields in algebraic geometry
14K05 Algebraic theory of abelian varieties
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