Chapter I

Projective modules

and vector bundles

The basic objects studied in algebraic K-theory are projective modules over

a ring and vector bundles over schemes. In this chapter we introduce the

cast of characters. Much of this information is standard, but it is collected

here for ease of reference in later chapters.

Here are a few running conventions we will use. The word ring will

always mean an associative ring with 1 (1 = 0). If R is a ring, the word

R-module will mean right R-module unless explicitly stated otherwise.

1. Free modules, GLn, and stably free modules

If R is a field or a division ring, then R-modules are called vector spaces.

Classical results in linear algebra state that every vector space has a basis

and that the rank (or dimension) of a vector space is independent of the

choice of basis. However, much of this fails for arbitrary rings.

As with vector spaces, a basis of an R-module M is a subset {ei}i∈I such

that every element of M can be expressed in a unique way as a finite sum

∑

eiri with ri ∈ R. We say that a module M is free if it has a basis. If

M has a fixed ordered basis, we call M a based free module and define the

rank of the based free module M to be the cardinality of its given basis.

Homomorphisms between based free modules are naturally identified with

matrices over R.

The canonical example of a based free module is

Rn

with the usual basis;

it consists of n-tuples of elements of R, or “column vectors” of length n.

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