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**The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory.**
*(English)*
Zbl 0715.14009

Algebraic \(K\)-theory: Connections with geometry and topology, Proc. Meet., Lake Louise/Alberta (Can.) 1987, NATO ASI Ser., Ser. C 279, 241-342 (1989).

[For the entire collection see Zbl 0685.00007.]

Let X be an appropriate scheme. The article gives a definition and systematic treatment of a new Grothendieck topology on X, called the completely decomposed topology. The topology is connected to several classical adelic constructions associated with group schemes, e.g., adele groups and adele class groups. The second goal is to construct a descent spectral sequence for the K-theory of coherent sheaves using the completely decomposed topology. This refines the Brown-Gersten spectral sequence.

The Brown-Gersten spectral sequence is an approximation to a conjecture of Quillen and Lichtenbaum. Let \(K_ n(X)\), respectively \(G_ n(X)\), be Quillen’s K-groups of the category of coherent locally free, respectively coherent, sheaves of \({\mathcal O}_ X\)-modules. Denote by \(\tilde K_ n^{et}\) resp. \(\tilde G_ n^{et}\) the sheafification of the corresponding presheaves sending Y to \(K_ n(Y)\), respectively \(G_ n(Y)\). The conjecture claims for regular X the existence of a spectral sequence converging to \(K_{q-p}(X,{\mathbb{Z}}/\ell {\mathbb{Z}})\) for sufficiently large q-p whose \(E_ 2\)-term is given in terms of étale cohomology by \(E_ 2^{p,q}=H^ p(X_{et},\tilde K_ q^{et}({\mathbb{Z}}/\ell {\mathbb{Z}}))\). The Brown-Gersten spectral sequence is based on the Zariski topology: \(E_ 2^{p,q}=H^ p(X_{zar},\tilde G_ q^{zar})\Rightarrow G_{q-p}(X)\) and similar for \({\mathbb{Z}}/\ell {\mathbb{Z}}\)-coefficients. If X is regular, one can replace G by K. The new spectral sequence of this article is based on the completely decomposed topology: \(E_ 2^{p,q}=H^ p(X_{cd},\tilde G_ q^{cd})\Rightarrow G_{q-p}(X).\) Since the completely decomposed topology is stronger than the Zariski topology and weaker than the étale topology, this spectral sequence refines the Brown-Gersten sequence and is a step towards the conjectured étale spectral sequence of Quillen and Lichtenbaum. The applicability of this new spectral sequence for calculations of \(K_ i(X,{\mathbb{Z}}/\ell {\mathbb{Z}})\) depends on understanding the K-theory of fields.

Let X be an appropriate scheme. The article gives a definition and systematic treatment of a new Grothendieck topology on X, called the completely decomposed topology. The topology is connected to several classical adelic constructions associated with group schemes, e.g., adele groups and adele class groups. The second goal is to construct a descent spectral sequence for the K-theory of coherent sheaves using the completely decomposed topology. This refines the Brown-Gersten spectral sequence.

The Brown-Gersten spectral sequence is an approximation to a conjecture of Quillen and Lichtenbaum. Let \(K_ n(X)\), respectively \(G_ n(X)\), be Quillen’s K-groups of the category of coherent locally free, respectively coherent, sheaves of \({\mathcal O}_ X\)-modules. Denote by \(\tilde K_ n^{et}\) resp. \(\tilde G_ n^{et}\) the sheafification of the corresponding presheaves sending Y to \(K_ n(Y)\), respectively \(G_ n(Y)\). The conjecture claims for regular X the existence of a spectral sequence converging to \(K_{q-p}(X,{\mathbb{Z}}/\ell {\mathbb{Z}})\) for sufficiently large q-p whose \(E_ 2\)-term is given in terms of étale cohomology by \(E_ 2^{p,q}=H^ p(X_{et},\tilde K_ q^{et}({\mathbb{Z}}/\ell {\mathbb{Z}}))\). The Brown-Gersten spectral sequence is based on the Zariski topology: \(E_ 2^{p,q}=H^ p(X_{zar},\tilde G_ q^{zar})\Rightarrow G_{q-p}(X)\) and similar for \({\mathbb{Z}}/\ell {\mathbb{Z}}\)-coefficients. If X is regular, one can replace G by K. The new spectral sequence of this article is based on the completely decomposed topology: \(E_ 2^{p,q}=H^ p(X_{cd},\tilde G_ q^{cd})\Rightarrow G_{q-p}(X).\) Since the completely decomposed topology is stronger than the Zariski topology and weaker than the étale topology, this spectral sequence refines the Brown-Gersten sequence and is a step towards the conjectured étale spectral sequence of Quillen and Lichtenbaum. The applicability of this new spectral sequence for calculations of \(K_ i(X,{\mathbb{Z}}/\ell {\mathbb{Z}})\) depends on understanding the K-theory of fields.

Reviewer: W.Lück

### MSC:

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

19E20 | Relations of \(K\)-theory with cohomology theories |