# zbMATH — the first resource for mathematics

Cartier-Dieudonné theory for Chow groups. (English) Zbl 0545.14013
Let k be a perfect field of positive characteristic, X a smooth projective variety over k and $${\mathcal C}$$ the category of augmented artinian local k-algebras. For $$A\in {\mathcal C}$$ let $${\mathcal K}^ s_{n,X\times A/X}$$ be the sub-sheaf of the sheaf of Quillen’s K-groups on $$X\times_{spec k}spec A$$ which is locally generated by Steinberg symbols with at least one factor $$\equiv 1 mod {\mathcal O}_ X\otimes_ k \max id A.$$ In this paper the cohomology groups of these sheaves are analysed; more precisely, one studies the functor $$H^*(X,{\mathcal K}^ s_{*,X\times /X}):{\mathcal C}\to$$ {bigraded abelian groups}.
It is shown that this functor admits a sort of Künneth decomposition with one factor solely depending on X and the other factor a functor on $${\mathcal C}$$ independent of X. This decomposition gives an analogue of Cartier’s formula relating a formal group to its Cartier-Dieudonné module. This result is remarkable because $$H^*(X,K^ s_{*,X\times /X})$$ is usually far from being pro-representable. The analogue of a Cartier-Dieudonné module for this functor turns out to be the $$E_ 1$$- term $$H^*(X,W\Omega^._ X)$$ of the slope spectral sequence for the crystalline cohomology of X [cf. L. Illusie and M. Raynaud, Publ. Math., Inst. Hautes Etud. Sci. 57, 73-212 (1983; Zbl 0538.14012)]. This result generalises the well-known connection between the formal Picard group, or more generally the Artin-Mazur groups [cf. M. Artin and B. Mazur, Ann. Sci. Éc. Norm. Super, IV. Ser. 10, 87- 132 (1977; Zbl 0351.14023)] and Witt vector cohomology. Because of Bloch’s formula $$CH^ n(X)=H^ n(X,{\mathcal K}_{n,X})$$ [cf. D. Quillen in Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)] we like to think of $$H^*(X,{\mathcal K}^ s_{*,X\times /X})$$ as the formal completion of the Chow group of codimension n cycles modulo rational equivalence on X, but it is not yet clear what our theorem about the structure of $$H^*(X,{\mathcal K}^ s_{*,X\times /X})$$ really means for the Chow groups. As a first step in understanding the connection we present a formal analogue of the Abel-Jacobi map $$CH_ 0(X)\to Alb(X).$$ As another step in understanding our result better we give, in case X is a K3- surface, a presentation for the groups $$H^ 2(X,{\mathcal K}_{2,X\times A/X})$$ in terms of generators and relations, which is an intriguing mixture of a formal group law and the well-known presentation for $$K_ 2$$. In the course of our analysis of $$H^*(X,{\mathcal K}^ s_{*,X\times /X})$$ we give a new K-theoretic construction of the De Rham-Witt complex.

##### MSC:
 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14C15 (Equivariant) Chow groups and rings; motives 14L05 Formal groups, $$p$$-divisible groups 14F30 $$p$$-adic cohomology, crystalline cohomology 14C99 Cycles and subschemes 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)
Full Text: