Preface

Algebraic K-theory has two components: the classical theory which centers

around the Grothendieck group K0 of a category and uses explicit algebraic

presentations and higher algebraic K-theory which requires topological or

homological machinery to define.

There are three basic versions of the Grothendieck group K0. One in-

volves the group completion construction and is used for projective mod-

ules over rings, vector bundles over compact spaces, and other symmetric

monoidal categories. Another adds relations for exact sequences and is used

for abelian categories as well as exact categories; this is the version first used

in algebraic geometry. A third adds relations for weak equivalences and is

used for categories of chain complexes and other categories with cofibrations

and weak equivalences (“Waldhausen categories”).

Similarly, there are four basic constructions for higher algebraic K-

theory: the +-construction (for rings), the group completion constructions

(for symmetric monoidal categories), Quillen’s Q-construction (for exact

categories), and Waldhausen’s wS. construction (for categories with cofi-

brations and weak equivalences). All these constructions give the same

K-theory of a ring but are useful in various distinct settings. These settings

fit together as in the table that follows.

All the constructions have one feature in common: some category C is

concocted from the given setup, and one defines a K-theory space associated

to the geometric realization BC of this category. The K-theory groups are

then the homotopy groups of the K-theory space. In the first chapter, we

introduce the basic cast of characters: projective modules and vector bundles

(over a topological space and over a scheme). Large segments of this chapter

will be familiar to many readers, but which segments are familiar will depend

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