Algebraic K-theory of schemes and spaces

This is a first course on the algebraic K-theory of schemes and spaces. Our goal is to cover the contents of two seminal papers in the subject, namely, Thomason's "Higher algebraic K-theory of schemes" and Waldhausen's "Algebraic K-theory of spaces". Time permitting, we will also treat additional topics, such as the relationship of K-theory to Chow groups via the Chern character map and the Grothendieck-Riemann-Roch theorem, or its relationship to (topological) Hochschild homology and stable homotopy via the cyclotomic trace map and the stable parametrized h-cobordism theorem. We will begin by defining the K-theory of a Waldhausen category via Waldhausen's S-construction. This specializes to produce the algebraic K-theory spectrum of a ring, a scheme, or a space, and we will study its relationship to Quillen's higher K-theory of an exact category, defined in terms of the Q-construction. Our first main goal will be to prove Waldhausen's additivity theorem, as well as the closely related fibration and approximation theorems. We will then specialize these results to the to the algebro-geometric world. In particular, we will define the (nonconnective) algebraic K-theory of a scheme X in terms of its derived category of perfect complexes and prove Thomason's results on localization. We will then show how this implies the Mayer-Vietoris sequence, and use this to obtain the projective bundle theorem and Bass' fundamental theorem. Our final goal will be to introduce and study the algebraic K-theory of spaces. Specifically, a connected topological space X determines a ring spectrum whose K-theory is the so-called A-theory of X; equivalently, A(X) is the K-theory of a certain Waldhausen category of retractive spaces over X. In case X is a manifold, the A-theory spectrum A(X) fits as the second term of a cofibration sequence in which the first term is the stable homotopy of X and the third term is the Whitehead space of X, a space closely related to the classifying space of the group of diffeomorphisms of X. Supplementary lectures on simplicial sets and homotopy theory will be given as necessary.