Cartier-Dieudonné theory for Chow groups.

*(English)*Zbl 0545.14013Let 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.

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) |