Tensor products in \(p\)-adic Hodge theory
There is a classical relation between the \(p\)-adic absolute value of the eigenvalues of Frobenius on crystalline cohomology and Hodge numbers for a variety in characteristic \(p\): ``The Newton polygon lies on or above the Hodge polygon''. For a variety in characteristic \(p\) with a lift to characteristic 0, Fontaine conjectured, and Faltings proved, a more precise statement: There is an inequality which relates the slope of Frobenius on any Frobenius-invariant subspace of the crystalline cohomology to the Hodge filtration, restricted to that subspace. A vector space over a \(p\)-adic field together with a \(\sigma\)-linear endomorphism and a filtration which satisfies this inequality is called a weakly admissible filtered isocrystal. The category of such objects is one possible \(p\)-adic analogue of the category of Hodge structures: in particular, it is an abelian category. We give a new proof of Faltings' theorem [\textit{G. Faltings} in: Proc. Internat. Congr. Math., ICM 94, 648-655 (1995; Zbl 0871.14010)] that the tensor product of weakly admissible filtered isocrystals over a \(p\)-adic field is weakly admissible. By a similar argument, we also prove a characterization of weakly admissible filtered isocrystals with \(G\)-structure in terms of geometric invariant theory, which was conjectured by Rapoport and Zink. Section 1 defines filtered isocrystals and explains how they arise geometrically. Sections 2 and 3 explain the ideas from geometric invariant theory which are used in the proof. Section 4 proves the tensor product theorem, and section 5 generalizes it to some bigger categories of filtered objects. Finally, sections 6-8 prove the characterization of weakly admissible filtered isocrystals with \(G\)-structure.
tensor product of weakly admissible filtered isocrystals
scientific article; zbMATH DE number 912121
scientific article