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MATH 634: Commutative
Algebra (Fall 2005)
Instructor:
Michael Stillman
Meeting
Time & Room
Prerequisites: Math 631 or equivalent.
Textbooks:
(1) Eisenbud: Commutative algebra (recommended)
(2) Greuel and Pfister: A Singular introduction to commutative algebra
(recommended).
This course is an introduction to commutative algebra. Commutative algebra
is a key ingredient of both algebraic number theory and algebraic geometry,
and is a lively area of research on its own.
Tentative list of topics:
- Groebner bases
- Ideals: localization, prime ideals, and operations on ideals
- Modules: graded rings and modules, syzygies, free resolution, tensor
products, and operations on modules
- Noether normalization and Hilbert's nullstellensatz, and several other
key concepts/theorems in commutative algebra: integral dependence, going
up/down and integral closure.
- Primary decomposition in Noetherian rings
- Discrete valutation rings
- Hilbert functions and dimension theory
- Tor and Ext, and flatness
- Some other topics in homological commutative algebra: Cohen-Macaulay
rings, depth and regular sequences
Throughout the entire course we will include many examples, often using
my computer algebra system Macaulay2 (written with Dan Grayson). Since
it is crucial to do mathematics in order to learn it, there will also
be regular homework assignments.
Last modified:
August 19, 2005
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