Place and time:
Tuesdays and Thursdays, 11:30am-1:00pm, in EH 3088.
Monday 2pm-3pm, Tuesday 9am-10am, and Thursday 1pm-2pm, in EH 4827. I can meet at other times by request.
, due Thursday 11/29 (updated 11pm 11/16)
, and some comments
Bonus worksheet on Tor
Bonus worksheet on normalization
, and a supplement
Bonus worksheet on Gröbner bases
The course is an introduction to commutative algebra, a subject that has
interactions with algebraic geometry, number theory, combinatorics, and several complex variables. The emphasis will be on Noetherian rings.
Topics will include the Noetherian property, integral extensions, Hilbert's Nullstellensatz, Noether normalization, localization, chains of prime ideals, Krull dimension, Artinian rings, normal Noetherian rings, flatness, primary decomposition, symbolic powers, Rees rings and the Artin-Rees Lemma, the Krull height theorem, and completion of local rings.
My plan is to motivate these topics by posing and solving some easily stated questions, such as:
- Given a finite set of symmetries of a polynomial ring, is there a finite set S of polynomials such that any polynomial fixed by the symmetries can be expressed in terms of elements of S?
- To what extent is a system of polynomial equations determined by its solution set?
- Is there an N(n) such that every system of polynomial equations in n variables can be replaced by a system of N(n) equations?
We will not follow any particular textbook, but recommended sources include:
The lecture notes for this class should not be used as a reference, but rather as an invitation to check out the sources listed above.
Commutative algebra software online interfaces
Math 593 and 594 or equivalent are prerequisites. If you are less than "very comfortable" with the definition of a module over a ring (and lack some guiding examples) and/or are less than "somewhat OK" with free modules, Hom of R-modules, and/or tensor products, I strongly encourage you to quickly review these notions. A friendly source is Dummit and Foote, 10.1-10.4.
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Math 631 is an introduction to Algebraic Geometry, our nearest neighbour.
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