Descent properties of homotopy \(K\)-theory. (English) Zbl 1079.19001
Let \(KH\) be Weibel’s homotopy invariant version of \(K\)-theory agreeing with \(K\)-theory for smooth schemes. In the paper under review the author shows that \(KH\) satisfies \(cdh\)-descent for essentially finite schemes \(X\) over a field \(F\) of characteristic zero. As a consequence we have a spectral sequence
\[
H^p(X,a_{cdh}K_{-q})\rightarrow KH_{-p-q}(X).
\]
Here \(a_{cdh}K_{-q}\) is the \(cdh\)-sheafification of the \(K\)-theory presheaf.
Reformulating the problem, the author proves that the associated presheaf \(\mathcal{KH}\) of spectra on \(Sch/k\) is locally fibrant in Jardine’s local \(cdh\)-model structure. Note that \(K\)-theory and \(KH\)-theory are locally equivalent. Letting \(\mathcal K^{cdh}\) be a common local fibrant replacement of \(\mathcal K\) and \(\mathcal{KH}\), the goal is to show that \(\mathcal{KH}(X)\to \mathcal K^{cdh}(X)\) is a weak equivalence. This is done in two steps.
First the equivalence is established for hypersurfaces by choosing a resolution of singularities and factoring it into blow-ups with regularly embedded centers and finite abstract blow-ups. Equivalence for the first of these is established in the paper under review, and equivalence for the latter has been shown by Weibel.
The second step in the proof of the theorem is reducing the general case to the hypersurface case, which is done by induction on the dimension. One needs only descent for closed covers and invariance under infinitesimal extensions.
In the last chapter some evidence for a conjecture of Weibel concerning negative \(K\)-theory is given.
The paper is based on the author’s thesis.
Reformulating the problem, the author proves that the associated presheaf \(\mathcal{KH}\) of spectra on \(Sch/k\) is locally fibrant in Jardine’s local \(cdh\)-model structure. Note that \(K\)-theory and \(KH\)-theory are locally equivalent. Letting \(\mathcal K^{cdh}\) be a common local fibrant replacement of \(\mathcal K\) and \(\mathcal{KH}\), the goal is to show that \(\mathcal{KH}(X)\to \mathcal K^{cdh}(X)\) is a weak equivalence. This is done in two steps.
First the equivalence is established for hypersurfaces by choosing a resolution of singularities and factoring it into blow-ups with regularly embedded centers and finite abstract blow-ups. Equivalence for the first of these is established in the paper under review, and equivalence for the latter has been shown by Weibel.
The second step in the proof of the theorem is reducing the general case to the hypersurface case, which is done by induction on the dimension. One needs only descent for closed covers and invariance under infinitesimal extensions.
In the last chapter some evidence for a conjecture of Weibel concerning negative \(K\)-theory is given.
The paper is based on the author’s thesis.
Reviewer: Bjørn Dundas (Bergen)
MSC:
| 19D35 | Negative \(K\)-theory, NK and Nil |
| 19E08 | \(K\)-theory of schemes |
| 14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |
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