Stable \(K\)-theory as a derived functor
The authors consider stable \(K\)-theory of a ring \(R\) as a functor \(K^ s_ *(R,-)\): \({\mathcal B} \to {\mathcal A}b\), where \({\mathcal B}\) is the abelian category of functors \(T:{\mathcal R}^{op}_ f \times {\mathcal R}_ f \to {\mathcal A}b\), and \({\mathcal R}_ f\) is the additive category of finitely generated, free \(R\)-modules. The category \({\mathcal B}\) has a set of projective generators, \(\{P^{n,m} | n, m \in \mathbb{N} \cup \{0\}\}\), so the situation is amenable for classical homological algebra. The authors' Conjecture A states that the groups \(K^ s_ i(R,P^{n,m})\) vanish for all \(n,m\) and all \(i > 0\). Using basic stability results of \textit{W. van der Kallen} [Homology stability for linear groups, Invent. Math. 60, 269-295 (1980; Zbl 0415.18012)], the authors verify the conjecture for any semi-simple ring and any commutative integral domain of finite Krull dimension. Furthermore, Conjecture A is shown to imply that \(K^ s_ i(R,-)\) is the \(i\)-th derived functor of \(K^ s_ 0(R,-)\). Motivated by the description of \(THH_ * (R,M)\) as the homology groups of the small category \({\mathcal R}_ f\) with coefficients in \(\Hom (-,M \otimes_ R-) \in {\mathcal B}\), the authors define topological Hochschild homology \(THH_ *(R,T) : = H_ *({\mathcal R}_ f,T)\), for any \(T \in {\mathcal B}\). There results a natural transformation \(\nu : K^ s_ * (R,- ) \to THH_ *(R,-)\). The rest of the paper deals with the question: In which cases is \(\nu\) an isomorphism? Basically, I find the approach of this paper quite attractive. However, at least in a couple of spots, more precision in statements and care in proofs would have made for easier reading. First of all, I object to the use (in theorem 2) of ``='' to indicate the existence of a (not even natural) isomorphism between the abelian groups in question. A better statement of theorem 2 appears to be that ``the spectral sequence in question collapses and involves no extension problem.'' In the proof of theorem 2, the authors do mention a fair amount of relevant facts, but the reading is hard, not only because of the sloppy formulation of the theorem, but also because the authors fail to mention that the spectral sequence behaves well viz-a-viz the long exact sequences induced by a short exact sequence of coefficient functors. In fact, if the authors had stated the theorem precisely and isolated the salient ``naturality'' of the spectral sequence clearly, some of the details actually given could probably have been safely left to the reader so that the overall length of the exposition would not have increased, while the reading would have been much easier. Next, I have a question about the orbit set, \([V]\), introduced at the beginning of section 1. Presumably, in many examples, the apparent dependence on the dimension \(i\) (which is never mentioned) is real. Therefore, a comment concerning the behaviour under stabilization would have been nice to have. As it is, the authors have chosen a wording which -- if taken literally -- implies that \([V]\) is independent of \(i\). Finally, I spotted a few misprints that are slightly disturbing: In theorem 1, ``\({\mathcal B}\):'' is missing. On page 250, statement (i) deals with \(E^ \infty_{p,q}\) rather than \(E^ 2_ \infty\). In lemma 1, an extra condition, \(K^ s_ i (R,P) = 0\) for \(i>0\), was intended.
scientific article; zbMATH DE number 701717
scientific article