Descent for \(K\)-theories
Let \(X\) be a quasi-separated and quasi-compact scheme. Let \(K(X), K^{TT}(X),\) and \(K^{B}(X)\) denote the algebraic, Thomason-Trobaugh and Bass \(K\)-theory of \(X\), respectively. The authors extend some descent theorems by Thomason to these kind of schemes. More concretely, given a scheme \(X\), denote by \(X_{\tau}\) its associated site, \(\widetilde{X}_{\tau}\) its topos, \(X_{\text{Zar}}\) its Zariski site, \(X_{\text{ét}}\) its étale site, \(X_{\text{cd}}\) its cd site, and \(X_{\text{rcd}}\) its restricted cd site. The first results are bounds for the cohomological (cd) and mod-\(\ell\) étale virtual cohomological \((\text{vcd}_\ell)\) dimension of some of these sites, the first result is: Let \(X\) be a quasi-compact and quasi-separated scheme. Then, \(\text{cd}(\widetilde{X}_{\text{rcd}})\leq \dim(X)\) and \(\text{vcd}_{\ell}(\widetilde{X}_{\text{ét}})\leq \dim(X)+ \sup\{ \text{vcd}_{\ell}(k(x))\}_{x\in X}\). The next results are the descent theorems, these compare TT and Bass \(K\)-theory for quasi-compact and quasi-separated schemes. They are obtained by comparing certain corresponding spectral sequences. In order to state the theorems, one needs the following terminology: Let \(K^{B}/\ell(\;)\) denote the mod-\(\ell\) Bass \(K\)-theory and \(L_{\mathcal{K}}K^{B}/\ell (\;)\) its Bousfield \(\mathcal{K}\) localization. Theorem: Let \(X\) be a quasi-compact and quasi-separated scheme. Assume that \(\dim(X)<\infty\), \(1/\ell\in \Gamma (X,\mathcal{O}_{X})\), and that the residue fields of \(X\) have finite and uniformly bounded mod-\(\ell\) virtual étale cohomological dimension. Then threre is a weak equivalence \(K^{B}/\ell(X)\to L_{\mathcal{K}}K^{B}/\ell (X)\) on \(\sup\{\text{vcd}_{\ell}(k(x))-2\}_{x\in X}\)-connected covers. The authors use their results to prove the following descent result. Theorem: Let \(X\) be a quasi-compact and quasi-separated scheme of finite Krull dimension. Let \(S\) be a set of primes such that for all \(\ell\in S\), \(1/\ell\in \Gamma(X,\mathcal{O}_{X})\) and all residue fields of \(X\) have finite uniformly bounded mod-\(\ell\) virtual étale cohomological dimension. Let \(-\otimes \mathbb{Z}_{(S)}\) be the localization which inverts all primes not in \(S\). Then on \(\sup\{\text{vcd}_{\ell}(k(x))-2\}\)-connected covers, there is a weak equivalence \[ K^{B}(X)\otimes\mathbb{Z}_{(S)}\to \mathbb{H}_{\text{ét}}(X,K^{B}(\;)\otimes \mathbb{Z}_{(S)}. \] There is also a weak equivalence \[ L_{\mathcal{K}}K^{B}(X)\otimes\mathbb{Z}_{(S)}\to \mathbb{H}_{\text{ét}}(X,L_{\mathcal{K}}K^{B}(\;)\otimes \mathbb{Z}_{(S)}. \]
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