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

IHP
Salle 314

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\$.