These notes were prepared for a series of ten lectures given at Regional Conference of the Conference Board of the Mathematical Sciences in June 1971. The bulk of the lectures were on projective dimensions of "very large" modules. Although the material in these notes is not new, there are several places where existing work has been simplified. For example, a commutative local nondomain of global dimension 3 is described without reference to analysis, and the dimension of a quotient field of a polynomial ring rather than a regular local ring is calculated. A derivation of Tor, one step at a time without the usual derived functor machinery, is also included.

Readership

Table of Contents

• Introduction

• General definitions, notations, examples
• Basic properties of projectives, injectives, flat modules, Hom and \(\otimes\)
• Basic commutative algebra

• Definitions of various dimensions, Ext, and Tor
• An alternative derivation of Tor and Ext
• Elementary applications
• Commutative algebra revisited
• Set theoretic propositions
• Not so elementary applications and counting theorems
• More counting
• Appendix. Introductory set theory
• Notation, definitions, basic axioms
• Cardinals, ordinals, and the axiom of choice
• Bibliographical notes