×

\(K\)-theory for triangulated categories. I(A): Homological functors. (English) Zbl 0906.19002

This article is the first of a series adressing the following problem: Can one define \(K\)-theory directly for derived categories? A motivation for this problem is Waldhausen’s theorem: Let \(F:{\mathcal E}\rightarrow {\mathcal F}\) be any exact functor of exact categories. Suppose the induced map \(D^b({\mathcal E})\rightarrow D^b({\mathcal F})\) is an equivalence of categories. Then the natural map between Quillen’s constructions \(Q({\mathcal E })\rightarrow Q({\mathcal F})\) is a homotopy equivalence [F. Waldhausen, “Algebraic \(K\)-theory of spaces”, Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006)].
The article is continued in [A. Neeman, “\(K\)-theory for triangulated categories. I(B): Homological functors”, Asian J. Math. 1, No. 3, 435-529 (1997; Zbl 0906.19003)]. We give separate accounts for the two parts of the paper: This review concerns the results of the I(A)-part of the article. They are listed below. One should point out that the style of the article is sometimes unorthodox. In particular, the introduction, which contains remarks like “I have largely forgotten what argument I had” (p. 340) is a bit surprising.
The first two sections deal with triangulated categories and list some general results and definitions. In particular, a commutative square is called Mayer-Vietoris (MV for short) if the corresponding sequence of maps \(X\rightarrow Y\oplus Y'\rightarrow Z\) is part of a triangle.
If \(\mathcal T\) is a triangulated category and \(\mathcal S\subset{\mathcal T}\) is an exact subcategory, \(\mathcal S\) has the structure of a bicategory. Horizontally and vertically it is simply \(\mathcal S\), while the distinguished squares are the MV squares. To this bicategory a bisimplicial set is associated (the definition of which requires the introduction of an “extra differential” in the definition of \((p,q)\)-simplices). This approach yields a \(K\)-theory, defined as the homotopy of the geometric realization of this bisimplicial set. There are other ways of viewing \(\mathcal S\) as a bicategory. The squares have always to be MV squares, but one may for example insist that all horizontal (or vertical) morphisms are mono or epi. Therefore various bicategories and canonical inclusions between them emerge. The theorem 3.7, which is a fundamental result, claims that these natural inclusions induce homotopy equivalences. Together with the results of section 5 (Waldhausen-style rigidifications), it enlightens many (multi)simplicial constructions in \(K\)-theory (Quillen’s, Waldhausen’s …) and makes clear why they all are equivalent.
The section 4 deals with homological functors from a triangulated category to an abelian category \(\mathcal A\). Let \(Gr^b({\mathcal A})\) be the category of “bounded graded” objects in \(\mathcal A\). It is naturally a bicategory with squares MV squares. This section introduces the theorem 4.8, which asserts that the natural inclusion between the bisimplicial sets associated respectively to \(\mathcal A\) and \(Gr^b({\mathcal A})\) (which are defined by the same kind of process as before) induces a homotopy equivalence. The proof of theorem 4.8 is continued in the second part I(B) of the paper.
These are the contents of the part I(A) of the article. One should insist on the fact that the results of section 3 and 5 on multisimplicial constructions in \(K\)-theory are of interest in their own.
Reviewer: F.Patras (Nice)

MSC:

19D06 \(Q\)- and plus-constructions
18E30 Derived categories, triangulated categories (MSC2010)
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
18G10 Resolutions; derived functors (category-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI