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 3.00, May 2, 2013).


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. Integrality
  7. Artinian rings
  8. Direct and inverse limits
  9. Tensor products
  10. Flatness
  11. Finitely generated projective modules
  12. Zariski's lemma and the Hilbert Nullstellsatz
  13. The spectrum of a ring
  14. Jacobson rings and max spectra
  15. Quasi-finite algebras and Zariski's main theorem
  16. Dimension theory for finitely generated k-algebras
  17. Primary decompositions
  18. Dedekind domains
  19. Dimension theory for noetherian rings
  20. Regular local rings
  21. 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, 2012). Revised, expanded. 95 pages.