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Modification of the Fontaine-Laffaille functor. (English. Russian original) Zbl 0731.14026
Math. USSR, Izv. 34, No. 3, 467-516 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 3, 451-497 (1989).
Let \(K_ 0\) be a finite extension of the p-adic numbers \({\mathbb{Q}}_ p\) with residue field of q elements. Let \(K=K_ 0\otimes W(k)\) be its maximal unramified extension, \(\bar K\) an algebraic closure of K, \(\Gamma =Gal(\bar K| K)\) the corresponding Galois group. Let A and \(A_ 0\) be the rings of integers of K and \(K_ 0\) respectively. Let MF be the category of filtered Dieudonné A-modules and \(M\Gamma\) the category of \(A_ 0\)-modules provided with a continuous linear action of \(\Gamma\). Then Fontaine and Lafaille constructed an important exact functor U: MF\(\to M\Gamma\); its importance lies in the fact that if \(K_ 0={\mathbb{Q}}_ p\) in dimensions \(<p\) it establishes a transformation from the crystalline cohomology (provided with the Hodge filtration) of a smooth proper scheme over W(k) to its étale cohomology (with its natural \(\Gamma\)-module structure).
In turn this was used to prove a generalized Shafarevich conjecture: \(\dim (H^ i(X\otimes {\mathbb{C}},\Omega^ j))=0\) for \(i\neq j\), \(i+j\leq 3\) for smooth proper schemes X over \({\mathbb{Z}}\). There are obstructions to generalizing these results and techniques directly. With a view of circumventing this the author in the present paper, constructs and discusses a modified Fontaine-Lafaille type functor to obtain a suitable functor \(\tilde U:\) MF\(\to \Phi M\Gamma^*\) where the latter category consists of triples \((H_ 0,H,i)\) with \(H_ 0,H\in M\Gamma\) and i an embedding (in \(M\Gamma\)). The functor coincides with U on the subcategory of unipotent objects of MF (when identifying \(M\Gamma\) with \(\{(0,H,0)\}\subset \Phi H\Gamma^*)\). Applications to smooth schemes over \({\mathbb{Z}}\) will appear in a future paper [cf. the author, Math. USSR, Izv. 35, No.3, 469-518 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No.6, 1135-1182 (1989; Zbl 0733.14008)].

14L05 Formal groups, \(p\)-divisible groups
14F30 \(p\)-adic cohomology, crystalline cohomology
18F99 Categories in geometry and topology
14F20 Étale and other Grothendieck topologies and (co)homologies
18E99 Categorical algebra
Zbl 0733.14008
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