Algebraic \(K\)-theory via binary complexes. (English) Zbl 1276.19003

The \(K\)-theory of an exact category was defined by D. Quillen [Lect. Notes Math. 341, 85–147 (1973; Zbl 0292.18004)] as the homotopy of a space (or a spectrum) constructed from the category. If the exact category is actually an additive category, there is an explicit description of the lower \(K\)-groups (in dimensions \(\leq 2\)). For a general exact category, a representation of the elements (or the generators) of \(K\)-groups has been a long standing problem. For the \(K_0\)-group there is an explicit description as the Grothendieck group of the exact category. For \(K_1\), the generators were given by A. Nenashev [J. Pure Appl. Algebra 131, No. 2, 195–212 (1998; Zbl 0923.19001)] and C. Sherman [\(K\)-Theory 14, No. 1, 1–22 (1998; Zbl 0901.19001)].
In the paper under review the author generalizes the above construction to give a presentation of the \(K_n\)-group of any exact category. In the paper, the \(K\)-theory spectrum is used. The construction is based on a generalization of the double exact sequences that appear in the previous references. First, it is pointed out that the category of bounded chain complexes with quasi-isomorphisms (maps that induce isomorphisms in homology) is a category with weak equivalences. For an exact category \(\mathcal{N}\), the author defines the binary chain complexes over \(\mathcal{N}\) to be bounded chain complexes with two differentials. The main technical result of the paper is the categorical description of the loop exact category of an exact category. So \({\Omega}{\mathcal{N}}\) is defined to be the arrow, in the category of exact categories, of the diagonal functor from the category of acyclic chain complexes to the category of acyclic binary chain complexes. In the definition, in both categories, the weak equivalences are isomorphism. It is proved that the \(K\)-spectrum of \({\Omega}{\mathcal{N}}\) is the connective part of the loops of the \(K\)-spectrum of \(\mathcal{N}\). That is done is three stages. First it is shown that the \(K\)-spectrum of \({\Omega}{\mathcal{N}}\) is equivalent to the loop of the relative \(K\)-spectrum of the diagonal functor between the same categories with weak equivalences being quasi-isomorphisms. In the second stage, it is shown that the \(K\)-spectrum of \({\Omega}{\mathcal{N}}\) is equivalent to the loops of the \(K\)-spectrum of the binary complexes that the first complex is acyclic with weak equivalences being quasi-isomorphisms. For the third result, it is shown that the last \(K\)-spectrum is equivalent to the \(K\)-spectrum of the category of chain complexes whose Euler characteristic vanishes. For those results, Waldhausen’s approximation and fibration theorem are used. The main result follows from Thomanson’s cofinality theorem.
The results mentioned above are proved first for exact categories that support long exact sequences. That means that \(\mathcal{N}\) admits a fully faithful exact functor into an abelian category \(\mathcal{A}\) so that the class of objects of \(\mathcal{A}\) isomorphic to objects of \(\mathcal{N}\) are closed under extensions and for acyclic chain complexes in \(\mathcal{A}\) whose objects are in \(\mathcal{N}\), the images of the differentials are in \(\mathcal{N}\). For the extension to the general case, the author uses Thomanson’s construction that adds images for the idempotent maps of an exact category and embeds it, as a full subcategory, to an exact category that supports long exact sequences. The two categories (and their loop categories) have naturally isomorphic \(K\)-spectra from the cofinality theorem.
Iterating the construction, the author constructs the exact category \({\Omega}^n{\mathcal{N}}\) whose \(K\)-spectrum is equivalent to the connective part of the \(n\)-loop spectrum of the \(K\)-spectrum of \(\mathcal{N}\). Using this result, he constructs a presentation of \(K_n{\mathcal{N}}\). The generators are binary multicomplexes of dimension \(n\) in \(\mathcal{N}\). The relations are coming from the exact sequences and the trivial binary multicomplexes (those that one of their components is in the image of the diagonal functor).


19D99 Higher algebraic \(K\)-theory
18E10 Abelian categories, Grothendieck categories
Full Text: DOI


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