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

vendredi 27 mars 2015, 14:00 → 15:30 Europe/Paris

Salle 314 (IHP)

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 $A_i:=A/p^{i}A$. The main result is a natural quasi-isogeny between the relative K-theory pro-spectrum "lim"$K(A_i,I{A}_i)$ and the cyclic pro-complex "lim"$CC(A_i,I{A}_i)$. 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 ${\oplus }_a{H}_{dR}^{2a-n-1}(X_E)/F^a$.