A spectrum level rank filtration in algebraic \(K\)-theory
Previously rank filtrations for the algebraic \(K\)-theory of a ring \(R\) were defined using \(BGl_ k(R)^ +\), the plus construction on the classifying space of the general linear group \(Gl_ k(R)\), or related concepts involving \(Gl_ k(R)\). Using Waldhausen's \(S_ \bullet\)- construction the author takes a slightly different approach by restricting to the subcategory of \(S_ \bullet{\mathcal F}(R)\) of diagrams involving only free \(R\)-modules of \(\text{rank}\leq k\), \({\mathcal F}(R)\) denoting the category of finitely generated free \(R\)-modules. For rings satisfying a certain condition (there should only exist injections \(R^ i \to R^ j\) with free cokernel) he arrives at a filtration of the \(K\)- theory spectrum \(\mathbf{K}R\) by subspectra \(F_ k\mathbf{K}R\) which exhaust \(\mathbf{K}R\). The previous approaches did not give spectra. The main part of the paper is an analysis of the layer \(F_ k\mathbf{K}R/F_{k-1}\text\textbf{K}R\) which turns out to be the homotopy orbit spectrum of a spectrum \({\mathcal D}(R^ k)\) with \(Gl_ k(R)\)-action. Filtering \({\mathcal D}(R^ k)\) by a poset filtration arising naturally from the \(S_ \bullet\)-construction the author finds \({\mathcal D}(R^ k)\) to be a suspension spectrum and identifies its homology as \(Gl_ k(R)\)-module as the homology of an explicitly given chain complex \(E_ *(k)\) of length \((2k-2)\). The spectral sequence of the homotopy orbit spectrum \[ E^ 1_{s,t} = H_ t(Gl_ k(R);E_ s(k)) \Rightarrow H_{s+t}(F_ k\mathbf{K}R/F_{k-1}\text\textbf{K}R) \] makes the layers of the poset filtration accessible to computations. The paper ends with such computations in low ranks. The author has provided a new and useful method for calculating \(K\)- groups (at least in low dimensions). Using this approach he could prove the vanishing of \(K_ 4(\mathbb{Z})\) [\(K_ 4(\mathbb{Z})\) is the trivial group, to appear in Annals of Mathematics].
scientific article; zbMATH DE number 167787