Ring Theory Appendix: Reference Coverage Map
Ring theory / appendix
Reference coverage map
This appendix records where the series uses Atiyah and Macdonald, *Introduction to Commutative Algebra*, as its commutative-algebra reference.
Coverage of Introduction to Commutative Algebra
| Topic in Introduction to Commutative Algebra | Used in this series |
|---|---|
| Chapter 1: rings and ring homomorphisms | 01, 09 |
| Chapter 1: ideals and quotient rings | 01, 03, 08 |
| Chapter 1: zero divisors, nilpotents, units | 01, 08 |
| Chapter 1: prime and maximal ideals | 02, 03 |
| Chapter 1: prime and maximal ideals as residue-field points | 02, 03, 08, 10 |
| Chapter 1: nilradical and radicals | 01, 08 |
| Chapter 1: extension and contraction of ideals | 02, 09 |
| Chapter 2: modules and homomorphisms | 01, 06 |
| Chapter 2: submodules and quotient modules | 01, 06, 07 |
| Chapter 2: finitely generated modules | 01, 07 |
| Chapter 2: exact sequences | 01, 06, 07 |
| Chapter 2: tensor products and extension of scalars | 01, 06, 09 |
| Chapter 3: rings and modules of fractions | 02, 05, 06 |
| Chapter 3: local properties | 02, 05, 06, 07 |
| Chapter 3: ideals in rings of fractions | 02, 03, 08 |
| Chapter 3: local rings as algebraic neighborhoods | 02, 05, 08, 11 |
| Chapter 6: chain conditions | 07 |
| Chapter 7: Noetherian rings | 07, 08 |
| Chapter 7: Hilbert basis theorem context | 07 |
| Chapter 8: Artin rings | Mentioned only indirectly through zero-dimensional local behavior |
| Chapter 9: DVRs and Dedekind domains | Not developed; relevant to future arithmetic geometry expansion |
| Chapter 10: completions | Not developed; relevant to future formal-local expansion; 11 uses only finite-jet intuition |
| Chapter 11: dimension theory | 08 |
| Chapter 11: Noetherian local rings and regular local rings | 08, with only introductory use |
| Local algebra behind tangent-space intuition | 08, 11, with only introductory use |
Supplementary expository bridges
| Theme | Used in this series |
|---|---|
| Residue-field value viewpoint for \(\mathbb C[x]\) and \(\mathbb Z\) | 02, 03, 05, 08, 10 |
| Calculus and Taylor-series analogy for local rings | 11 |
| Smooth versus real algebraic examples \(S^2\) and \(T^2\) | 11 |
Deliberate scope choices
The series focuses on the path to affine schemes and coherent sheaves. It does not attempt to reproduce all of Atiyah and Macdonald. Primary decomposition, integral dependence, valuation rings, completions, Dedekind domains, and full dimension theory are acknowledged as later commutative-algebra topics rather than prerequisites for the first construction of affine schemes.
Suggested later expansions
- Primary decomposition and associated primes.
- Integral extensions, going-up, and normalization.
- Dedekind domains and arithmetic curves.
- Completion of local rings and formal neighborhoods.
- Regular local rings and smoothness.
- Projective schemes and Proj.
- Cohomology of coherent sheaves beyond the affine case.