The K-theory presheaf of spectra, by J. F. Jardine

This expository paper displays a construction of the algebraic K-theory presheaf of spectra, which starts with a method of functorially associating a symmetric spectrum K(M) to an exact category M. The symmetric spectrum K(M) is defined via well known techniques of Waldhausen. The categorical coherence problem implicit in defining the K-theory presheaf on the category of S-schemes is solved (as it has by other authors) by using big site vector bundles in place of ordinary vector bundles. Some applications are displayed: these include a Galois cohomological descent spectral sequence for the etale K-theory of a scheme (where the Galois group is the Grothendieck fundamental group), and the Morel-Voevodsky description of Thomason-Trobaugh K-theory as Nisnevich K-theory in non-negative degrees. There is also a spectrum-level description of Voevodsky's periocity operator for Nisnevich K-theory.

J. F. Jardine <>