Algebraic Structures Autumn 581D: Homological algebra

Instructor: Julia Pevtsova

Place: Padelford Hall, C-401

Time: 9:30-10:20, MWF

Office Hours: by appointment (that is, I am happy to talk between classes, just need an advance notice)

Course Description.  An introductory course on homological algebra. 

Topics:

·         Chain complexes, homotopies, homology and long exact sequence in homology

·         Resolutions, derived functors, Ext and Tor. Koszul complexes

·         Group (co)homology

·         Triangulated and derived categories

·         Spectral sequences

 

Grading system. Based on homework and class presentations.

 

Textbook.  C. Weibel,  An introduction to homological algebra (on reserve at the Math Library)

 

Other references.

General/comprehensive homological algebra texts:

1.      J. Rotman, An introduction to homological algebra, electronic version (UW restricted)

2.      H. Cartan, S Eilenberg, Homological algebra (even though outdated, this is  a classic where the foundations of the subject were laid out)

3.      S. MacLane, Homology

Group cohomology

1.      K. Brown, Cohomology of groups

2.      L. Evens, Cohomology of groups

Triangulated categories

1.      A. Neeman, Triangulated categories

2.      S. Gel’fand, Yu. Manin, Methods of homological algebra

3.      M. Hovey, J. Palmieri, N. Strickland, Axiomatic stable homotopy theory

4.      P. May, The axioms for triangulated categories

Also useful: S. MacLane, Categories for the Working Mathematician

 

Homework.

·         Homework 1 (pdf, tex, MyDefns.tex)

·         Homework 2 (pdf, tex)

·         Homework 3 (pdf, tex)

 

Presentation topics (1 or 2 people per topic):

1.      1lim (3.5 in Weibel) – Alys & Rebecca

2.      A short introduction to classifying spaces (when we do group cohomology) – Lorenzo & Sid

3.      The category K(R-mod) is triangulated – Yannick

4.      The stable module category for a Frobenius algebra R is triangulated – Cody & Kevin, Notes in .pdf

5.      The homotopy category of spectra – Reid & Riley

6.      Galois Cohomology - Bharath

 

Presentation schedule:

·         Wednesday, Nov. 28, in class           The triangulated category K(R-mod)

·         Monday, Dec 3rd, 4:45pm, in the lounge       The stable module category

·         Friday, Dec. 8th, in class        The homotopy category of spectra

·         Friday, Dec. 8th, afternoon time TBA, in the lounge            Galois cohomology