On determinant functors and $K$-theory

Details On determinant functors and $K$-theory On determinant functors and $K$-theory

2010-06-28

arxiv.org/abs/1006.5399v1

Mathematics

K-Theory and Homology

71 pages

Scientific paper

In this paper we introduce a new approach to determinant functors which allows us to extend Deligne's determinant functors for exact categories to Waldhausen categories, (strongly) triangulated categories, and derivators. We construct universal determinant functors in all cases by original methods which are interesting even for the known cases. Moreover, we show that the target of each universal determinant functor computes the corresponding $K$-theory in dimensions $0$ and $1$. As applications, we answer open questions by Maltsiniotis and Neeman on the $K$-theory of (strongly) triangulated categories and a question of Grothendieck to Knudsen on determinant functors. We also prove additivity and localization theorems for low-dimensional $K$-theory and obtain generators and (some) relations for various $K_{1}$-groups.

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