Holmstrom Determinants

Something in the work of Faltings/Gillet/Soule I think???

http://mathoverflow.net/questions/68218/is-there-an-algebraic-construction-of-the-quillen-determinant-line-bundle

arXiv:1006.5399 On determinant functors and $K$-theory from arXiv Front: math.CT by Fernando Muro, Andrew Tonks, Malte Witte 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.

nLab page on Determinants