Some filtrations on higher \(K\)-theory and related invariants
Let \(X\) be a scheme. The author defines groups \(KV^ j_ m(X)\) which can be thought of as the Karoubi-Villamayor theory of modules having codimenson \(\geq j\), and also groups \(KV_ m^{j/j+1}(X)\). This is done by lifting Bloch's higher Chow construction [\textit{S. Bloch}, Adv. Math. 61, 267-304 (1986; Zbl 0608.14004)] from abelian groups to topological spectra. The \(KV\) groups are the homotopy groups of simplicial spectra \(K^ j(X,\bullet)\) (respectively \(K^{j/j+1}(X,\bullet))\). There is a spectral sequence \(E_ 1^{pq}=KV^{p/p+1}_{-p-q}(X,n)\Rightarrow K_{-p-q}(X)\) \((K_ n\) being Quillen's higher \(K\)-functor) which leads to a filtration on higher \(K\)-theory. Additional invariants \(KH_ m^{j/j+1}(X,n)\) are defined which are the \(E^ 2\) terms of spectral sequences \(E^ 2_{p,q}=KH_ q^{j/j+1}(X,p)\Rightarrow KV^{j/j+1}_{p+q}(X)\). Under additional hypotheses there are spectral sequences \('E_ 1^{p,q}=KH_{r-p}^{p/p+1}(X,-q)\Rightarrow H^{p+q}(X,{\mathcal K}_ r)\) (where \({\mathcal K}_ r\) is the sheafification in the Zariski topology of the functor \(K_ r)\). These various spectral sequences are applied to the case \(X=\text{Spec}(k)\), \(k\) a field, in order to compute some of the \(KV\) and \(KH\) groups of a field. For example \(KH_ 3^{0/1}(X,0)\cong K_ 3(k)^{\text{ind}}\) and the \(KV^ j_ 3(X)\) \((0\leq j\leq 3)\) are described in terms of Bloch's higher Chow groups and the image of \(K^ M_ 3(k)\) (Milnor's \(K\)-group) in \(K_ 3(k)\). The paper concludes with a speculative discussion of how these invariants might relate to motivic cohomology.
scientific article; zbMATH DE number 130403
scientific article