A Primer of Commutative Algebra - J.S. Milne   Top
Expository Notes
A Primer of Commutative Algebra
Motives---Grothendieck's Dream
What is a Shimura Variety?
Introduction to Shimura Varieties
Shimura Varieties and Moduli
Tannakian Categories
The Work of Tate
pdf (current version 4.01, August 15, 2014).


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.


  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


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.