MAT 544: Commutative and Homological Algebra

TuTh    1:15pm- 2:35pm         Physics P122

 

Homework assignments

Homework 1 (due Sep. 10): Atiyah-Macdonald, Ch.2, #2, #8, #14, #16, #17; Ch.3, #6, #9

Homework 2 (due Sep. 22): Atiyah-Macdonald, Ch.6, #1i, #2, #9; Ch.7, #1, #4, #8, #11

Homework 3 (due Oct. 1): see this PDF file

Homework 4 (due Oct. 13): see this PDF file

Homework 5 (due Oct. 29): see this PDF file

Homework 6 (due Nov. 12): see this PDF file

Homework 7 (due Dec. 1): Atiyah-Macdonald, Ch.10, #3, #4, #9

Lectures

Here is a list of topics and recommended reading; [AM] is Atiyah-Macdonald; [Eis] is Eisenbud; [Ma] is Matsumura; [We] is Weibel.

 

     

Date	Topic	Reference
Aug 23	Review of rings, ideals, modules	[AM] Ch.1-2
Aug 25 Aug 30	Tensor product, localization	[AM] Ch.2-3
Sep 1	Exact sequences, flatness of localization	[AM] Ch.3
Sep 6	Spectrum of a ring	[AM] Ch.1 exercises
Sep 8	Noetherian rings	[AM], p.74-76, 80-81
Sep 13	Associated primes, definition, and examples	[Eis] p.87-92
Sep 15	Associated primes, main theorem	[Eis] p.90-94
Sep 20	Primary decomposition	[AM], p.50-51, 82-84
Sep 22	Nullstellensatz	[AM], p.81-82, 85
Sep 27	Non-noetherian examples, artinian rings	[AM], Ch.8
Sep 29	Artinian rings	[AM], Ch.8
Oct 4	Local rings, integral dependence	[AM], p.21-22, 59
Oct 6	Integral dependence	[AM], p.59-62
Oct 11 break	Integral dependence, integral closure	[AM], p.61-63
Oct 13	Dedekind domains, DVRs	[AM], p.93-95
Oct 18	Fractional ideals, unique factorization	[AM], p.96-98
Oct 20	Categories and functors	[We], p.417-424, 429-431
Oct 25	Adjoint functors, injective and projective modules	[We], p.33-35, 38-39
Oct 27	Injective modules, chain complexes, Tor	[We], p.38-39, 1-4, 15-18
Nov 1	Tor, long exact sequence in homology	[We], p.10-14, 36, 53
Nov 3	Symmetry of Tor, checking for flatness
Nov 8	Topologies and completions	[AM], p.100-105
Nov 10	Graded rings and modules, Artin-Rees lemma	[AM], p.105-108
Nov 15	Group project
Nov 17	Group project
Nov 22	Group project
Nov 24 Break
Nov29	Group project
Dec 1

About the course

Summary. An introduction to the techniques of commutative and homological algebra useful in algebra, algebraic geometry, number theory, and related fields. Review of rings and modules, tensor products and localization. Spectrum of prime ideals, Noetherian and Artinian rings and modules, completion, dimension theory, local rings, discrete valuation rings and Dedekind domains, integral dependence. Chain complexes, projective and injective resolutions, examples of derived functors (Ext and Tor), basic category theory (adjoint functors, natural transformations, limits and colimits), abelian categories. Here is a more detailed syllabus (including suggested reading).

 

Grading. Grades will be based on weekly homework assignments, class participation, and on your contribution to a group project at the end of the semester. There is no final exam.

 

Homework. Most homework assignments will come from the book Introduction to Commutative Algebra by Atiyah and Macdonald. Here is a link to a scanned copy.

 

Time and location

We meet on Tuesday and Thursday    1:15pm- 2:35pm         Physics P122.  My office is Math Tower 4–109; office hours will be held via Zoom on Fridays from 2:00pm to 4:00pm.

 

Policy Statements

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