Commutative Algebra
Program
- Ideals in commutative rings.
- Spectrum of a ring.
- Zariski topology.
- Radicals.
- Modules.
- Tensor product.
- Localization.
- Noetherian and Artinian rings.
- Primary decomposition.
- Support.
- Algebraic extensions,
Noether’s normalization theorem
and Hilbert Nullstellensatz.
- Integral extensions,
“going-up”
and “going down” theorems.
- Discrete valutation rings.
- Invertible Ideals.
- Completion,
Artin-Rees’ lemma,
Krull’s theorem,
Hensel’s theorem.
- Dimension theory.
Basic bibliography:
- Michael F. Atiyah, Ian G. Macdonald:
Introduction to Commutative Algebra.
Addison-Wesley, 1994.
- Allen Altman, Steven Kleiman:
A Term of Commutative Algebra
- James S. Milne:
A Primer of Commutative Algebra,
v4.03a (March 23, 2020), 113 pages. Available at www.jmilne.org/math
- Hideyuki Matsumura: Commutative Algebra,
Benjamin-Cummings Pub Co, 1980.
- Christian Peskine:
An algebraic introduction to Complex Projective Geometry: 1. Commutative Algebra.
Cambridge University Press, 2009.
Software:
CoCoA,
SAGE.