Cycleclasses for algebraic De Rham cohomology and crystalline cohomology

Abstract

For schemes which are smooth over a regular base scheme we establish the existence of cycle class maps with values in the corresponding algebraic De Rham cohomology. These maps have all the properties one expects, i.e. they are compatible with flat morphisms, base change and Künneth morphisms. Moreover, they are homotopy invariant with respect to affine bundles and in the few cases, where such cycle classes have been constructed previously (notably by Berthelot and Hartshorne), they coincide with those classes. Likewise, our cycle maps behave well with respect to intersection theory in the sense that they pass to rational equivalence and, when the base is the spectrum of a field and the considered schemes are smooth and quasi-projective, they respect the multiplicative structure of the Chow ring which is given either by K-theory via the Bloch-Quillen isomorphism or by using the refined Gysin morphisms of Fulton and Mac Pherson. Finally, it is shown how the cycle classes for crystalline cohomology constructed by Michel Gros using purity theorems for logarithmic Hodge-Witt cohomology can be obtained from our cycle maps.