Alexander Beilinson "Relative continuous K-theory and cyclic homology"

Alexander Beilinson (Chicago)
Relative continuous K-theory and cyclic homology

Let $A$ be a $p$-adic ring, $I$ its two sided ideal such that $p$-adic topology on $A$ equals $I$-adic one; set $A_i :=A/p^iA$. The main result is a natural quasi-isogeny between the relative K-theory pro-spectrum "lim"$K(A_i,IA_i)$ and the cyclic pro-complex "lim"$CC(A_i,IA_i)$. This is a $p$-adic version of the classical isomorphism of Goodwillie (to be recalled in the first half of the talk).
A geometric application (which is a generalization of a theorem of Bloch-Esnault-Kerz): Let $X$ be a proper scheme over the ring of integers of a $p$-adic field E such that the generic fiber $X_E$ is smooth, and $Y$ be its subscheme whose support equals the close fiber. Then the projective limit of relative non-connective K-groups $K_n^B (X/p^i,Y)$ identifies naturally, after being tensored by $\mathbb Q$, with Hodge-truncated de Rham cohomology $\oplus_a H_{dR}^{2a-n-1}(X_E)/F^a$.