Algebraic Preliminaries

This preliminary chapter is devoted to the following three subjects, which together allows us to review a great part of the prerequisites, and to add some more specialized knowledge:

(a)

a very simple presentation of the notion of universal property, with many examples; Sections

1.2

to

1.4

are devoted to this subject.

(b)

additional information about categories of modules; Sections

1.5

to

1.9

contain reminders about exact sequences, usual functors, projective modules and changes of basic rings; but in

1.8

finitely presented modules are treated with more detail.

(c)

a self-contained presentation of rings and modules of fractions, localization and globalization; although this material is already well treated in the existing literature, a concise exposition of the exact part that is here actually useful should prevent beginners from wandering in too-specialized topics.

Complete knowledge of

all

this chapter is not indispensable, because precise references will always be given when the most difficult or specialized results are needed in the following chapters.