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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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