Weigth zero motivic cohomology and the general linear group of a simplicial ring
The author compares the weight zero parts of two filtrations on the K-theory of a regular ring \(R\) all of whose residue fields are infinite. Both filtrations have been conjectured to lead to a motivic Atyah-Hirzebruch spectral sequence. The first filtration is called \(C^\bullet\) and was introduced in [\textit{S. E. Landsburg}, K-Theory 6, No. 5, 431-455 (1992; Zbl 0774.14006)]. The other filtration is called \(W^\bullet\) and was originally suggested by Goodwillie and Lichtenbaum. The point of this paper is to show that the weight zero parts agree. This also corrects a mistake in Landsburg's paper and shows that the \(C^\bullet\) filtration is actually better behaved than that paper indicates. In the computations the author exploits that one meets simplicial rings that satisfy the `primitive criterion' introduced by the reviewer [\textit{W. van der Kallen}, Ann. Sci. Éc. Norm. Supér., IV. Ser. 10, 473-515 (1977; Zbl 0393.18012)]. Recall that this criterion guarantees there are plenty of units.
scientific article; zbMATH DE number 1435992