Hypercohomology spectra and Thomason’s descent theorem

but with the convergence only valid in sufficiently high degrees. Here the coefficient sheaves are Tate twists of the `-adic integers, and are to be interpreted as zero if q is odd. Throughout this paper, etale cohomology is continuous etale cohomology [19], and the indicated abutment of the spectral sequence consists of the homotopy groups of the Bousfield `-adic completion of the spectrum KX, not the naive `-adic completion of the K-groups. In a remarkable paper [42], Thomason proved the Lichtenbaum-Quillen conjectures for a certain localized form of K-theory so-called “Bott-periodic” K-theory. The first step was the development of an elaborate theory of hypercohomology spectra H·(X; E) associated to etale presheaves of spectra E or more generally, to presheaves of spectra on a Grothendieck site. These hypercohomology spectra are by their very construction naturally