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Towards non-Abelian \(p\)-adic Hodge theory in the good reduction case. (English) Zbl 1213.14002

Mem. Am. Math. Soc. 990, v, 157 p. (2011).
The paper under review is about the comparison between étale and crystalline cohomology, for a smooth and proper scheme over a discrete valuation ring of mixed characteristic (“Fontaine theory”). In fact, slightly more generally the logarithmic theory with a simple normal crossing divisor is included but we ignore this for simplicity. The theory defines a fully faithful correspondence between certain étale \(p\)-adic sheaves on the generic fibre, and crystalline objects (filtered Frobenius isocrystals). The objects which have a partner on the other side are usually called admissible.
The author starts with a Tannakian category of semisimple admissible objects, and considers the bigger Tannakian category of all admissible objects which have a composition theories whose members lie in the original category. For example if we start with trivial representations we obtain unipotent sheaves. Choosing a base-point in our scheme defines fibre functions, and the inclusion of the two Tannakian categories corresponds to a surjection of algebraic groups. Its kernel is (on the étale side) a certain Malcev completion. The main result of the paper states that the two kernels (étale and crystalline) correspond to each other under the étale/crystalline comparison on the base.
The proof uses heavy machinery like modul categories, derived functors etc. For the layman the essence of the matter seems to be the following:
a) By considering objects trivialised at the base-point one eliminates \(H^0\)’s, and the \(H^1\)’s classify extensions and not just isomorphy class of extensions.
b) By degeneration of Hodge spectral sequences (for the elements of the smaller Tannakian category) filtered extensions of crystalline objects are classified by \(F^0H^1\).
c) Extensions of admissible objects are admissible.
d) The comparison respects connecting differentials in long exact sequences. Namely the bigger Tannakian category is filtered by considering the minimal length of a filtration with subquotients in the smaller. Objects of degree \(n+1\) are obtained as extensions of objects of degrees \(1\) and \(n\). There exist universal extensions both on the étale and crystalline side and they correspond. Thus the universal pro objects representing the two fibre functor correspond as well.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14G20 Local ground fields in algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
14-XX Algebraic geometry
11-XX Number theory
14F30 \(p\)-adic cohomology, crystalline cohomology
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