Date | Topics covered |
Monday January 13 |
Introduction |
Wednesday January 15 |
Review category theory and algebra |
Friday January 17 |
Review cont. |
Monday January 20 |
NO CLASS |
Wednesday
January 22 |
Chain complexes Complexes of R-modules, operations on chain complexes, long exact sequences. |
Friday January
24 |
Chain homotopies. |
Monday January 27 |
Mapping cones and mapping cylinders. |
Wednesday January 29 |
Mapping cones and mapping cylinders cont. |
Friday January 31 |
More on abelian categories. |
Monday
February 3 |
Derived Functors Delta functors, projective resolutions, injective resolutions. |
Wednesday
February 5 |
Left derived functors, right
derived functors. |
Friday
February 7 |
Adjoint functors and
left/right exactness. |
Monday February 10 |
Balancing Tor and Ext. |
Wednesday February 12 |
Derived
categories (preview). |
Friday
February 14 |
Tor and Ext Tor for abelian groups, Tor and flatness. |
Monday
February 17 |
Ext for nice rings. |
Wednesday
February 19 |
Ext and extensions. |
Friday
February 21 |
Universal coefficients
theorem |
Monday February 24 |
Cup product |
Wednesday February 26 |
Homological
dimension Dimensions, change of rings theorems. |
Friday February 28 |
Local rings. |
Monday
March 3 |
Koszul complexes. |
Wednesday
March 5 |
Local cohomology. |
Friday
March 7 |
Cohen--Macaulay rings. |
Monday March 10 |
Cohen--Macaulay rings. |
Wednesday March 12 |
Spectral
sequences Introduction, terminology. |
Friday March 14 |
The Leray--Serre spectral sequence. |
Monday
March 17 |
The spectral sequence of a
filtration. |
Wednesday
March 19 |
Convergence. |
Friday
March 21 |
Spectral sequences of a double
complex. |
Week of
March 24-March 28 |
SPRING BREAK |
Monday March 31 |
Hypercohomology (derived categories preview). |
Wednesday April 2 |
Grothendieck Spectral Sequences, applications. |
Friday April 4 |
Exact couples. |
Monday
April 7 |
The derived category Introduction, categories of complexes. |
Wednesday
April 9 |
Triangulated categories. |
Friday
April 11 |
Triangulated categories
continued. |
Monday April 14 |
Localization and the calculus of fractions. |
Wednesday April 16 |
The derived category. |
Friday April 18 | Derived functors. |
Monday
April 21 |
The total tensor product. |
Wednesday
April 23 |
Ext and RHom. |
Friday
April 25 |
Replacing spectral
sequences. |
Monday April 28 |
Applications |
Wednesday April 30 |
Applications |
Friday May 2 |
Applications |
Wednesday May 7 |
FINAL
EXAM 1:30 PM -4 PM |