$p$-Adic Deformation of Motivic Chow Groups

Abstract

For a smooth projective scheme $Y$ over $W(k)$ we consider an element in the motivic Chow group of the reduction $Y_m$ over the truncated Witt ring $W_m(k)$ and give a “Hodge” criterion – using the crystalline cycle class in relative crystalline cohomology – for the element to lift to the continuous Chow group of the associated $p$-adic formal scheme $Y_{\bullet }$. The result extends previous work of Bloch–Esnault–Kerz on the $p$-adic variational Hodge conjecture to a relative setting. In the course of the proof we derive two new results on the relative de Rham–Witt complex and its Nygaard filtration, and work with a relative version of syntomic complexes to define relative motivic complexes for a smooth lifting of $Y_m$ over the ind-scheme $\mathrm{Spec}{W}_{\bullet }(W_m(k))$.