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Cyclic homology, cdh-cohomology and negative $$K$$-theory. (English) Zbl 1191.19003
Let $$X$$ be a $$d$$-dimensional scheme, essentially of finite type over a field $$F$$ of characteristic zero. Proven is that $$K_m (X) = 0$$ for $$m < -d$$, where $$K$$ denotes (negative) algebraic K-theory. Furthermore, $$X$$ is shown to be $$K_{-d}$$-regular, where given a contravariant functor $$\mathcal{F}$$ on schemes, a scheme $$X$$ is called $$\mathcal{F}$$-regular if $${\mathcal{F}}(X) \to {\mathcal{F}}(X \times {\mathbb{A}}^r)$$ is an isomorphism for all $$r \geq 0$$. Letting cdh denote Voevodsky’s cdh-topology [V. Voevodsky, “Homotopy theory of simplicial sheaves in completely decomposable topologies”, J. Pure Appl. Algebra 214, No. 8, 1399–1406 (2010; Zbl 1194.55020)], the authors of the current paper prove the main theorem in the following steps:
[1.] $$K_{-d}(X) \simeq H^d_{\text{cdh}}(X; \, {\mathbb{Z}})$$;
[2.] $$H^d_{\text{Zar}}(X; \, {\mathcal{O}}_X) \to H^d_{\text{cdh}}(X; \, {\mathcal{O}}_X)$$ is surjective;
[3.] If $$X$$ is smooth, then $$H^n_{\text{Zar}}(X; \, {\mathcal{O}}_X) \simeq H^n_{\text{cdh}}(X; \, {\mathcal{O}}_X)$$ for all $$n$$.
Note that in characteristic zero, every scheme is locally smooth for the cdh-topology.
Let {HC}$$(X)$$ denote the cyclic homology complex associated with the mixed complex {C}$$(X)$$. In particular, {HC} is a presheaf of complexes on Sch/$$F$$ (schemes essentially of finite type over $$F$$). Let $${\mathbb{H}}_{\text{cdh}} (X; \, \xi)$$ be a fibrant replacement of the presheaf $$\xi$$ (in a suitable model structure). For a presheaf of spectra $$\xi$$ on Sch/$$F$$, $$\widetilde{C}_j$$ is the cofiber of the map $\xi \to \xi (\, \underline{\;\;} \times {\mathbb{A}}^j).$ Since $$\widetilde{C}_j(\xi)$$ is a direct factor of $$\xi (\, \underline{\;\;} \times {\mathbb{A}}^j)$$, the functor $$\xi \to \widetilde{C}_j \xi$$ preserves homotopy fibration sequences. Finally, let $${\mathcal{K}}(X)$$ denote the non-connective $$K$$-theory spectrum of perfect complexes on $$X$$. A key theorem in the paper under review is that there is an objectwise homotopy fibration sequence
$\widetilde{C}_j {\mathbf{HC}} \to {\mathbb{H}}_{\text{cdh}}(\,\underline{\;\;}\, ;\, \widetilde{C}_j {\mathbf{HC}}) \to \widetilde{C}_j{\mathcal{K}}.$
A useful tool in moving between $$K$$-theory and cyclic homology is Cortiñas’ infinitesimal $$K$$-theory [G. Cortiñas, J. Reine Angew. Math. 503, 129–160 (1998; Zbl 0908.19004)], which in the present paper is proven to satisfy descent for the cdh-topology. Recall that a presheaf $$\xi$$ of spectra on Sch/$$F$$ satisfies cdh-descent if $$\xi$$ satisfies the Mayer-Vietoris property for all elementary Nisnevich squares and for all abstract blow-up squares in Sch/$$F$$.

##### MSC:
 19D35 Negative $$K$$-theory, NK and Nil 19E08 $$K$$-theory of schemes 19D55 $$K$$-theory and homology; cyclic homology and cohomology 55U35 Abstract and axiomatic homotopy theory in algebraic topology
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