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MATH 8030, Spring 2017

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Snake Lemma

Brief Syllabus

The course text is Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994.

I hope to cover most of Chapters 1, 2, and 4, the first half of 5, and portions of 3, 6, 7, and 8.

Nominally, grades will be based on homework (presented in class and / or turned in for grading), class attendance, and class participation. If you want to be evaluated in a different way, please email me a proposal.

Office hours (Boyd 601B): Drop-in MWF after 10:00. Exceptions: Algebra Seminar M 3:30-4:30; research meeting W 1:30-2:45; lunch sometime during 11:30-1:00. 

Homework

Do as many of the problems as you have time for. Be prepared to discuss / present them in class on the "due date." Handing in written solutions is optional if you participate in the class discussion.

Assignment Due Date Problems
Asst. 1 Fri. Jan. 20 1.1.2, 1.1.4, 1.1.6 or 1.1.7 (your choice), 1.2.3, Verify at least two additional pieces of the Snake Lemma, 1.4.3, 1.4.5 (at least two parts; there's a typo in part 1.: "chain homotopy equivalence" should be "chain homotopic").
Asst. 2 Fri. Feb. 3 1.3.3, 1.4.2, 1.5.1, 1.5.2.
(1) Prove that any direct summand of a free R-module is projective.
(2) Prove that Hom_R(M , –) is left exact (without looking at the proof of Prop. 1.6.8; see Definition 1.6.6).
(3) Prove that P is projective iff Hom_R(P , –) is exact (without looking at the proof of Lemma 2.2.3).
(4) Calculate Ext_Z(Z/p,Z/q) for distinct primes p and q.
Asst. 3 Fri. Feb. 17 2.3.2 or 2.3.5, 2.4.2, 2.4.3, 2.5.2, 2.6.4.
(1) Show that "abelianization" is left adjoint to inclusion Ab -> Group.
Asst. 4 Fri. Mar. 3 2.7.3, 3.2.1, 3.2.2, 3.4.1.
(1) Prove Theorem 2.6.10 part 1.
(2) Check the missing details of the proof of Theorem 3.4.3.
Asst. 5 Fri. Mar. 24 6.1.1, 6.1.2, 6.1.3, 5.1.1 (see also 5.2.1; note correction on Weibel's website), 5.1.2, 5.1.3.
Asst. 6 Fri. Apr. 14 5.7.1, 5.7.2, 5.7.3.
(1) Verify the remark following Definition 5.7.3.
(2) Verify the claims about the "I" spectral sequence in Proposition 5.7.6.
(3) Use the Lyndon-Hochschild-Serre spectral sequence to compute the homology of Z/n from the short exact sequence  0 -> nZ -> Z -> Z/n -> 0.

Links

Many corrections to Weibel's book: http://www.math.rutgers.edu/~weibel/Hbook-corrections.html