Math 676: Commutative Algebra.
Commutative rings, modules, tensor products, localization, primary decomposition, Noetherian and Artinian rings, application to algebraic geometry and algebraic number theory.
Desired Learning Outcomes
Minimal learning outcomes
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.
- Commutative rings and ideals
- Tensor products
- Primary decomposition
- Integral dependence
- Noetherian and Artinian rings
- Dedekind domains and discrete valuation rings
- Applications to algebraic geometry
Possible textbooks for this course include (but are not limited to):
- Atiyah & Macdonald, Introduction to Commutative Algebra
- Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry
Possible additional topics are:
- Dimension theory