# zbMATH — the first resource for mathematics

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)].

##### MSC:
 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
Full Text: