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Relative continuous $$K$$-theory and cyclic homology. (English) Zbl 1337.19002
To state the main result of the paper, let $$A$$ be a unital, associative, $$p$$-adic ring, $$p$$ a prime, such that the ideal of $$p$$-torsion elements in $$A$$ is annihilated by $$p^n$$ for some $$n$$. Let $$A = \underset{\longleftarrow}{\lim} A_i$$, $$A_i := A/{p^i}$$ and suppose that $$A_1$$ has finite stable rank. Let $$I$$ be a two-sided ideal of $$A$$ such that the $$I$$-adic topology coincides with the $$p$$-adic topology. Let $$I_i$$ be the image of $$I$$ in $$A_i$$. Let $$K(A, \, I)^{\widehat \;}$$ be the relative $$K$$-theory pro-spectrum of the projective system of rings $$A_i$$ and ideals $$I_i$$. Let $$CC(A)^{\widehat \;}$$ denote the cyclic chain pro-complex. Proven is that there is a natural homotopy equivalence of pro-spectra up to quasi-isogeny: $K(A, \, I)^{\widehat \;}_{\mathbb{Q}} \longrightarrow CC(A)^{\widehat \;}[1]_{\mathbb{Q}}.$ This result represents a continuous version of Goodwillie’s rational isomorphism [T. G. Goodwillie, Ann. Math. (2) 124, 347–402 (1986; Zbl 0627.18004)] between relative K-theory and relative cyclic homology with respect to a nilpotent two-sided ideal. Recall that a map $$f: X \to Y$$ (in an additive category) is an isogeny if there is a map $$g: Y \to X$$ such that both $$f \circ g = n \cdot {\mathbf{1}}_Y$$ and $$g \circ f = n \cdot {\mathbf{1}}_X$$ for some integer $$n$$, $$n \neq 0$$. The proof of the main result uses ideas from the Loday-Quillen-Tsygan isomorphism [J.-L. Loday, Cyclic homology. 2nd ed. Grundlehren der Mathematischen Wissenschaften. 301. Berlin: Springer (1998; Zbl 0885.18007)], relating Lie-algebra homology to cyclic homology as well as results of Lazard, Volodin and Suslin.
As an application, let $$E$$ be a $$p$$-adic field, $$O_E$$ the ring of integers and let $$X$$ be a proper $$O_E$$-scheme with smooth generic fiber $$X_E$$. Let $$Y \subseteq X$$ be any closed subscheme whose support equals the closed fiber. Set $$X_i = X \otimes {\mathbb{Z}}/{p^i}$$. Then the nonconnective K-theory groups $$K^B_{n-1}(X_i, \, Y)$$ are related (rationally) to the deRham cohomology groups $$\sum_{a} H_{\text{dR}}^{2a-n}/F^a$$, where $$F^*$$ denotes the Hodge filtration.

##### MSC:
 19D35 Negative $$K$$-theory, NK and Nil
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