Continuous \(K\)-theory
For a scheme \(X\) the continuous \(K\)-theory \(K_n^{\text{cts}} (X; \mathbb{Z}_\ell)\) is defined and its properties are derived. This parallels Jannsen's construction of continuous étale cohomology as a derived functor theory. For a scheme \(X\) let \(BQP (X)\) denote the classifying space of Quillen's \(Q\)-construction on the category \(P(X)\) of finite rank locally free \({\mathcal O}_X\)-modules. Let \(Y^n_{\ell^k}\) be the cofiber of \(S^{n - 1} @>\ell^k>> S^{n - 1} \to Y^n_{\ell^k}\) and write \(Y^n_{\ell^\infty}=\varprojlim Y^n_{\ell^k}\) (where the limit is taken over \(k)\). Then, by definition, \(K_n^{\text{cts}} (X; \mathbb{Z}_\ell) :=[Y^{n + 1}_{\ell^\infty}, BQP (X)]\) for \(n \geq 1\). \(K_n^{\text{cts}} (X; \mathbb{Z}_\ell)\) fits in an exact sequence: \[ 0 \to \varprojlim{^1 K_{n+ 1}}(X; \mathbb{Z}/\ell^k) \to K_n^{\text{cts}} (X; \mathbb{Z}_\ell) \to K_n (X; \mathbb{Z}_\ell) \to 0, \] where \(K_n (X; \mathbb{Z}_\ell)=\varprojlim K_n (X; \mathbb{Z}/\ell^k)\). One has a Bockstein exact sequence relating the various \(K_n^{\text{cts}} (X; \mathbb{Z}_\ell)\) and \(K_n (X; \mathbb{Z}/\ell^k)\). As a first result one has, under the assumption that \(K_n (X)_\ell\) contains no non-trivial divisible subgroup for \(n > 0\), isomorphisms \(K_n (X) [\ell^k] @>\sim>> K_n^{\text{cts}} (X; \mathbb{Z}_\ell) [\ell^k]\) for \(n > 0\) and \(K_n (X)/ \ell^k @>\sim>> K_n^{\text{cts}} (X; \mathbb{Z}_\ell)/ \ell^k\) for \(n > 1\). As a corollary one obtains a canonical isomorphism \(c :K_n (X)_\ell @>\sim>> K_n^{\text{cts}} (X; \mathbb{Z}_\ell)_\ell\) which restricts to an isomorphism on the divisible elements of both groups. For number fields one gets more precise results on divisibility. Several special cases are described: (i) For \(X=\text{Spec} \mathbb{F}_q\) one has \(K_n^{\text{cts}} (\mathbb{F}_q; \mathbb{Z}_\ell)=K_n (\mathbb{F}_q) \otimes \mathbb{Z}_\ell\); (ii) similarly for the ring of integers in a number field; (iii) for a field \(F\) with \(\mu_{\ell^\infty} \subset F\), \(K_2^{\text{cts}} (F; \mathbb{Z}_\ell)=K_2 (F; \mathbb{Z}_\ell)\); (iv) for a local \(F\) such that the residue characteristic is not equal to \(\ell\), and \(n > 1\), one has \(K_n^{\text{cts}} (F; \mathbb{Z}_\ell)=K_n (F; \mathbb{Z}_\ell)=K_n (F)_\ell\). One can relate \(K_n^{ \text{cts}} (X; \mathbb{Z}_\ell)\) and the derived functor formalism to étale \(K\)-theory \(K_n^{\text{ét}} (X)\). In particular, for a number field \(F\), one shows the isomorphisms: (i) \(K_n^{\text{cts}} (F; \mathbb{Q}_\ell)@>\sim>> K_n^{\text{ét}} (F) \otimes \mathbb{Q}_\ell\), and (ii) \(K_n (F; \mathbb{Z}_\ell) \otimes \mathbb{Q}_\ell @>\sim>> K_n^{\text{ét}}(F; \mathbb{Z}_\ell) \otimes \mathbb{Q}_\ell\), where \(K_n^{\text{ét}} (F; \mathbb{Z}_\ell)=\varprojlim K_n^{\text{ét}} (X; \mathbb{Z}/ \ell^k)\). Still in the spirit of the \(\lim - \lim^1\)-formalism one can prove for a number field a \(\lim^1\)-result which at the lim-level is equivalent to the Quillen-Lichtenbaum conjecture: \(\varprojlim^1 K_n (F; \mathbb{Z}/ \ell^k) @>\sim>> \varprojlim^1 K_n^{\text{ét}} (F; \mathbb{Z}/ \ell^k)\). As a final result it is shown that for a number field \(F\) and \(n > 1\) there is a natural isomorphism \(K_n^{\text{cts}} (F; \mathbb{Z}_\ell)/ \text{tors} @>\sim>> K_n^{\text{ét}} (F; \mathbb{Z}_\ell)/ \text{tors}\).
scientific article; zbMATH DE number 800397
scientific article