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      H. Cartan, S. Eilenberg, Homological algebra (even though outdated, this is a classic where the foundations of the subject were laid out)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

 

Proof of the Snake Lemma

 

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