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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