Algebraic Structures 509A: Homological
algebra; Spring 2019
Instructor: Julia Pevtsova
Place: Padelford Hall, C-36
Time: 10:30-11:20, 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 or open topic depending on the class interests
Grading system. Based on homework and class presentations.
Textbook. C. Weibel,
An introduction to homological algebra
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 H. Cartan, S. Eilenberg, Homological algebra (even though outdated, this is a classic where the foundations of the subject were laid out)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
Algebraic topology/homotopy theory
1. A.
Hatcher, Algebraic topology, electronic version
2. E. Spanier, Algebraic topology
Group cohomology
3. K.
Brown, Cohomology of groups
4. 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, due Friday April 19
Homework 2, due Friday, May 17 tex
Presentations:
·
Lim 1
. Isaac Boekelheide May 6, in
class
·
Classifying
spaces. Olivia Borghi and Adam Kapilow May
24, in class
·
Lie
Algebra cohomology: Curtiss Lyman and Cody Tipton May 17, 5pm.
·
Brown
Representability. Nico Courts, May
31, in class
·
Crush
course on spectral sequences. Sean Griffin, June 3, in class
·
Galois
Cohomology. Thomas Browning and Thomas Carr, June 5, in class
·
Etale
cohomology. Kristine Hampton and Tejas Devanur, June 7, in class
Outlines due on Friday, May 17.
Write ups/summaries/extensive discussions WITH references are due after the presentation has been given
Announcements:
There will be NO classes on Friday, April 12th and Monday, April 15th. Independent reading will be assigned.
Afternoon homework discussion and presentation on Lie algebra cohomology– May 17, 5pm. Hand-out with prerequisites for the Lie algebra presentation
Presentations notes:
Lim^1, Isaac Boekelheide
Brown Representability, Nico Courts
Classifying spaces, Olivia Borghi and Adam Kapilow
Lie Algebra cohomology, Curtiss Lyman and Cody Tipton
Galois Cohomology, Thomas Browning and Thomas Carr
Sheaf Cohomology, Kristine Hampton