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Global applications of relative \((\varphi,\Gamma)\)-modules. I. (English) Zbl 1163.11051

Berger, Laurent (ed.) et al., Représentation \(p\)-adiques de groupes \(p\)-adiques I. Représentations galoisiennes et \((\varphi, \Gamma)\)-modules. Paris: Société Mathématique de France (ISBN 978-2-85629-256-3/pbk). Astérisque 319, 339-419 (2008); erratum Astérisque 330, 543-554 (2010).
Let \(K\) be a local field containing \({\mathbb Q}_p\). The category of \(p\)-adic representations of \(\text{Gal}(\overline{K}/K)\) is equivalent to Fontaine’s category of étale \((\phi,\Gamma)\)-modules: objects in this latter category are, in principle, much more accesible than \(p\)-adic representations of \(\text{Gal}(\overline{K}/K)\), in particular lend themselves to explicit calculations. For example, the Galois cohomology of a \(p\)-adic representations \(T\) of \(\text{Gal}(\overline{K}/K)\) can effectively be computed from its assciated \((\phi,\Gamma)\)-module.
On the other hand, the ‘geometric’ invariants and features of \(T\), e.g. if \(T\) comes from the \(p\)-adic étale cohomology of a smooth \(K\)-variety, are not easily visible from its \((\phi,\Gamma)\)-module. This lead Fontaine to speculate, already along with his construction of the said equivalence of categories, that there might or should be a geometric or relative version of \((\phi,\Gamma)\)-modules which would allow the assignement of ‘relative’-\((\phi,\Gamma)\)-modules to small open pieces of (models of) \(K\)-varieties, and would ideally have sheaf properties.
A construction of such ‘relative’-\((\phi,\Gamma)\)-modules had been achieved in earlier work by Andreatta, and the present paper continues this program. There are three groups of results:
I) Local results: For rings \(R\) (respectively extensions \(\overline{R}\) of \(R\)) which describe, very roughly, small affine/affinoid pieces in smooth \({\mathcal O}_K\)-varieties (resp. generically étale coverings of such), it is proved that the Galois cohomology of a \(\text{Gal}(\overline{K}/K)\)-module \(M\) can be calculated in terms of a certain cochain complex with values in terms of the generalized \((\phi,\Gamma)\)-module associated with \(M\). This generalizes the computation of the Galois cohomology of \(p\)-adic representations \(T\) of \(\text{Gal}(\overline{K}/K)\) mentioned above.
II) Global results: Let \(X\) be a smooth, proper, geometrically irreducible scheme of finite type over \({\mathcal O}_K\) and let \({\mathbb L}\) denote a locally constant étale sheaf of \({\mathbb Z}/p^s{\mathbb Z}\)-modules (for some \(s\geq1\)) on the generic fibre \(X_K\) of \(K\). Let \({\mathfrak X}\) denote the formal completion of \(X\) along the special fibre. The authors construct a spectral sequence which computes the étale cohomology of \({\mathbb L}\) in terms of local, on \({\mathfrak X}\), relative \((\phi,\Gamma)\)-modules attached to \({\mathbb L}\).
III) Finally, constructions related to relative-\((\phi,\Gamma)\)-modules lead to certain Fontaine-sheaves (author’s terminology) on \(X\), and variants, but also new aspects of Faltings’s approach to \(p\)-adic Hodge theory are discussed, and actual and potential applications of these constructions are indicated.
For the entire collection see [Zbl 1156.14002].

MSC:

11G99 Arithmetic algebraic geometry (Diophantine geometry)
14F20 Étale and other Grothendieck topologies and (co)homologies
14F30 \(p\)-adic cohomology, crystalline cohomology
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