pdf (current version 4.03, March 23, 2020).

Abstract

These notes prove the basic theorems in commutative algebra required for algebraic number theory, algebraic geometry, and algebraic groups. They assume only a knowledge of the algebra usually taught in advanced undergraduate or first-year graduate courses.

Contents

1. Rings and algebras
2. Ideals
3. Noetherian rings
4. Unique factorization
5. Rings of fractions
6. Integral dependence
7. The going-up and going-down theorems
8. Noether's normalization theorem
9. Direct and inverse limits
10. Tensor products
11. Flatness
12. Finitely generated projective modules
13. Zariski's lemma and the Hilbert Nullstellsatz
14. The spectrum of a ring
15. Jacobson rings and max spectra
16. Artinian rings
17. Quasi-finite algebras and Zariski's main theorem
18. Dimension theory for finitely generated k-algebras
19. Primary decompositions
20. Dedekind domains
21. Dimension theory for noetherian rings
22. Regular local rings
23. Flatness and fibres
24. Completions

History

v1.00 (January 1, 2009). First version on the web. 51 pages. pdf (old version 1.00).
v2.00 (April 5, 2009). Revised and completed. 64 pages. pdf (old version 2.00).
v2.10 (May 30, 2009). Many improvements to the exposition (thanks to Shu Otsuka). 65 pages. pdf (old version 2.10).
v2.20 (April 10, 2010). Added section on finitely generated projective modules; minor fixes.74 pages. pdf (old version 2.20).
v2.21 (April 27, 2010). Minor fixes. 74 pages. pdf (old version 2.21).
v2.22 (March 30, 2011). Minor changes. 75 pages.
v2.23 (April 29, 2012). Corrected; minor improvements. 76 pages. pdf (old version 2.23).
v3.00 (May 2, 2013). Revised, expanded. Added proof of ZMT. 95 pages.pdf (old version 3.00).
v4.00 (August 1, 2014). Revised, expanded. 109 pages.
v4.01 (August 15, 2014). Minor fixes.pdf (old version 4.01).
v4.02 (March 18, 2017). Minor improvements, corrections.pdf (old version 4.02).
v4.03 (March 24, 2020). Minor fixes and improvements to the exposition.