** Spring 2011 **

** Professor: Dr. Hal Schenck**

324 Illini Hall

Office hours: following class, or by appointment.

Phone: 333-2229.

E-mail: schenck@math.uiuc.edu.

** Meeting times/rooms **

T/R 1230-150, AH 445

**Course Description**.
How can we understand a complicated object? The answer is to break
it into simpler pieces; for example, break a representation or a
module into a direct sum of irreducible components. If a module B is
already irreducible, this does not work, but we can often include
B in a short exact sequence 0 -> A -> B -> C -> 0. This is the
starting point for homological algebra. In this class we will cover

1. Snake lemma, homology, long exact sequence in homology. (Week 1, Chapter 1)

2. Projective and injective modules and resolutions, delta and derived functors. (Week 2, Chapter 2).

3. Ext and Tor. (Week 3, Chapter 3).

4. Local and Cech cohomology. (Week 4, Chapter 4).

5. Spectral sequences, techniques of computation, Serre spectral sequence,
Grothendieck spectral sequence of composite functors. (Weeks 5-7, Chapter 5).

6. Group cohomology, bar resolution. (Week 8, Chapter 6).

7. Time permitting: Derived categories, Gysin sequence,
Kunneth formula, universal coefficient theorem, Eilenberg-Moore
spectral sequence.

**Text.**

Charles Weibel: An introduction to homological algebra.

Useful supplemental material:

Hal Schenck: Notes on cohomology and spectral sequences, https://faculty.math.illinois.edu/~schenck/tapp.pdf.

Joseph J. Rotman. Introduction to Homological algebra.

Allen Hatcher. Spectral sequences in algebraic topology, http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html.

Spanier. Algebraic topology.

Brown. Cohomology of groups.

John McCleary. A user's guide to spectral sequences.

Saunders Mac Lane. Homology.

**Grading.** Your grade will be determined by
class participation and homework scores, problems will be assigned
in class and collected every three weeks.

HW 1: 1.3.3, 1.3.6, 2.1.2, 2.4.3 (due 2/15)

HW 2: 2.7.1, 3.2.1, 3.2.2, 3.3.1, A.4.7 and Eisenbud A.3.27: Prove the
multiplication on Yoneda Ext is well defined and associative. (due 3/8)

HW 3: 5.1.3, 5.2.2. For a homology spectral sequence, find a formula for d_3 (hint: snake lemma), and do Exercises 3.1-3.4 in my notes on spectral sequences. Extra credit problem: write out a careful proof of the Gysin sequence (5.3.7)--this is just mimicing the proof of the Wang sequence.(due 4/5)

**Academic Integrity** I encourage group work on
the homework problems. This does not include copying each others
solutions.

**Copying Course Materials:** All printed
hand-outs and web-materials are protected by US Copyright Laws. No
multiple copies can be made without written permission by the
instructor.

Updated 3/28/11 (hks).