A Primer of Commutative Algebra - J.S. Milne Top

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

Errata

pdf (current version 4.01, August 15, 2014).
### 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

### 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.

- Rings and algebras
- Ideals
- Noetherian rings
- Unique factorization
- Rings of fractions
- Integral dependence
- The going-up and going-down theorems
- Noether's normalization theorem
- Direct and inverse limits
- Tensor products
- Flatness
- Finitely generated projective modules
- Zariski's lemma and the Hilbert Nullstellsatz
- The spectrum of a ring
- Jacobson rings and max spectra
- Artinian rings
- Quasi-finite algebras and Zariski's main theorem
- Dimension theory for finitely generated
*k*-algebras - Primary decompositions
- Dedekind domains
- Dimension theory for noetherian rings
- Regular local rings
- Flatness and fibres
- Completions

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.