Generalised sheaf cohomology theories

This paper is an expanded version of notes for a set of lectures given at the Isaac Newton Institute for Mathematical Sciences during a NATO ASI Workshop entitled “Homotopy Theory of Geometric Categories” on September 23 and 24, 2002. This workshop was part of a program entitled New Contexts in Stable Homotopy Theory that was held at the Institute during the fall of 2002. The intent for the lectures was to present some of the basic features of the homotopy theory of simplicial presheaves and the stable homotopy theory of presheaves of spectra, and then display their use in applications. A general outline of these theories forms the subject of Sections 1 and 2 of this paper. There has been some renewed interest in equivariant stable categories for profinite groups of late, and the main features of that theory have been descibed here in Sections 3 and 4. I wanted to stress the calculational aspects of that theory as well as display the basic results. This is done in the course of presenting an outline of the proof of Thomason’s descent theorem for Bott periodic algebraic K-theory, which appears in Section 5. The outline of the Thomason theorem that is presented here is a stripped down version of the proof appearing in [21], with all of the hard bits (ie. the coherence issues) carefully swept under the rug. Also, the proof works as stated only for good schemes and at good primes. The other cases, which are much more complicated to discuss, have been treated in detail elsewhere, particularly in Thomason’s original paper [40] and the Thomason-Trobaugh paper [41]. One should also look at the commentary given by Mitchell in [32].