\(K\)-theory for triangulated categories. II: The subtlety of the theory and potential pitfalls
This article is the third of a series intended to define directly \(K\)-theory groups for triangulated categories. In particular, for the derived category of an abelian category \(\mathcal A\), one should recover Quillen's usual \(K\)-theory of \(\mathcal A\). The series began with \textit{A. Neeman} [I(A): ibid. 1, No. 2, 330-417 (1997; Zbl 0906.19002); I(B): ibid. 1, No. 3, 435-529 (1997; Zbl 0906.19003)]. The author introduces the paper as follows: ``The real purpose of this article is to warn the unwary beginner of the pitfalls of the theory. The article is only really of interest to the expert seeking to improve the results''. In other terms: this is a highly technical paper, which faces and solves deep problems. Regrettably it is written and organized in an unorthodox way. Section 1 deals with generalities on the (multi)simplicial sets involved in the definition of the author's \(K\)-theory groups and the homotopies between them. Section 2 discusses the simplicial sets themselves and ``is mostly intended to serve as a warning of potential pitfalls in the theory''. At last, section 3 deals with the main problem: the comparison of the \(K\)-theory of the derived category of an abelian category \(\mathcal A\) with the usual \(K\)-theory of \(\mathcal A\). Stronger statements on the \(K\)-theory of a small triangulated category with a \(t\)-structure will be proved in the following paper of the series.
scientific article; zbMATH DE number 1240914