Homological Dimensions of Modules

B. Osofsky9780821816622, 0-8218-1662-4

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.

---

Table of contents :
Homological dimensions of modules……Page 1
Table of Contents……Page 5
Introduction……Page 6
1. General definitions, notations, examples……Page 8
2. Basic properties of projectives, injectives. flat modules, Hom, and ⊗……Page 22
3. Basic commutative algebra……Page 30
1. Definitions of various dimension, Ext, and Tor……Page 35
2. An alternative derivation of Tor and Ext……Page 43
3. Elementary applications……Page 48
4. Commutative algebra revisited……Page 56
5…….Page 61
6. Not so elementary applications and counting theorems……Page 62
7. More counting……Page 67
1. Notations, definitions and basic axioms……Page 78
2. Cardinals, ordinals, and the axiom of choice……Page 87
Bibliographic notes……Page 93