Homological Methods

The class meets MWF at 10:10 in LGRT 1322.

Professor: Jenia Tevelev

Office Hours:

E-mail: tevelev(at)math.umass.edu

Evolving Lecture Notes:

Complexes. Long exact sequence. Jan 23.
Categories and functors. Simplicial homology. Singular homology. Jan 25.
Functoriality of singular homology. De Rham cohomology. Jan 27.
Free, projective, and injective resolutions. Jan 30.
Exact functors. Categories with enough injectives. Adjoint functors. Feb 1.
R-mod has enough injectives. Quasi-isomorphism and homotopy. Feb 3.
Homotopy invariance of singular homology and de Rham cohomology. Feb 8.
Dolbeaut cohomology and del-bar Poincare Lemma. Feb 10.
Koszul complex. Associated primes. Feb 13.
Regular sequences. Feb 15.
Mapping cone. Feb 17.
Inductive description of the Koszul complex. Feb 22.
Derived functors. Feb 24.
δ-functors. Feb 27.
Tor. Ext. Feb 29.
Ext1 and extensions. Mar 2.
Spectral sequence. Mar 5.
Spectral sequence of a filtered complex. Mar 7.
Spectral sequence of a double complex. Mar 9.
Ext and Tor using the second argument.
Fiber bundles. Homotopy groups. Mar 12.
Leray spectral sequence. Mar 14.
Monodromy. Mar 16.
Leray-Hirsch Theorem. Kunneth formula. Mar 26.
Sheaves. Ringed spaces. Mar 28.
Stalks. Sheafification. Mar 30.
Abelian categories. Categories of sheaves and sheaves of OX-modules. Apr 2.
Cohomology of sheaves. Apr 4.
De Rham resolution. Dolbeaut resolution. Apr 6.
Čech complex. Apr 9.
Interpretations of H1. Picard group. Apr 13.
Derived Categories - I (by Rina Anno). April 18.
Derived Categories - II (by Rina Anno). April 20.
Cap product, Poincare duality, Serre duality. April 23, 25, 27, 30

Homework 1.
Homework 2.
Homework 3.
Homework 4.
Homework 5.
Homework 6.