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Day & Time: Tuesday and Friday
9:30am — 10:50am
Location: Goldsmith 0226
Instructor:
Joel Bellaiche
Office hours: Tuesdays, 3pm — 4pm
Grade: 40% exercises in class, 40% personal project, 20% final exam
Textbooks: For references, I will essentially use Commutative Algebra (with a view toward algebraic geometry) by David Eisenbud. This is very long, very complete, very detailed book with a lot of explanations and motivations. This book is on reserve at the main library and on sale at the campus bookstore. Other textbooks include Commutative Algebra, and Commutative Ring Theory by Matsumura, which are quite the opposite in style: short, direct, concise; Attiyah-Macdonald's Introduction to commutative algebra; the classic Commutative Algebra of Zariski and Samuel; and of course, Bourbaki.
| Class |
Topic |
Suggested Reading |
Exercises |
| Jan 17 |
Introduction. History. |
Chapter 1 of Eisenbud; Weil's Number-Theory and Algebraic Geometry |
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| Jan 20 |
Basic Notions: rings, fields, algebras; ideals; prime and maximal ideals; Noetherian rings |
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Exercises 1 |
| Jan 24 |
Basic Notions |
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| Jan 27 |
The radical of a ring and of an ideal. Reduced rings. |
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| Jan 31 |
Spectrum and Zariski topology, |
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Exercises 2 |
| Feb 3 |
Spectrum and Zariski topology. The Nullstellensatz |
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| Feb 7 |
The Nullstellensatz and consequences. Beginning of Modules |
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| Feb 10 |
Morphisms of modules. Noetherian modules and noetherian rings. Operations on submodules. |
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| Feb 14 |
Operations on Modules: direct sum, product, Hom |
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| Feb 17 |
Operations on Modules: tensor product. |
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Exercises 3 |
| Feb 17 |
Operations on Modules: tensor product. |
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| Feb 28 |
Tensor product of algebras. The Cayley-Hamilton theorem |
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| Mar 2 |
Integral elements and integral closure.. |
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Exercises 4 |
| Mar 6 |
The nullstellensatz for general fields. |
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| Mar 9 |
Rings of fractions |
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| Mar 13 |
No class |
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| Mar 16 |
Localization. Local rings, Nakayama's lemma |
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Exercises 5 |
| Mar 20 |
Localization, flatness |
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| Mar 24 |
Flatness: criterion with ideals |
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| Mar 27 |
Flatness: criterion with relations. Flat modules over a local rings. |
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Exercises 6 |
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