Relative continuous \(K\)-theory and cyclic homology.

*(English)*Zbl 1337.19002To 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.

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.

Reviewer: Jerry M. Lodder (Las Cruces)

##### MSC:

19D35 | Negative \(K\)-theory, NK and Nil |