Tensor products in \(p\)-adic Hodge theory. (English) Zbl 0873.14019

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 [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.


14F30 \(p\)-adic cohomology, crystalline cohomology
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14G20 Local ground fields in algebraic geometry


Zbl 0871.14010
Full Text: DOI


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