SYLLABUS AND
HOMEWORK

MATH 6290

Homological Algebra

MATH 6290

Homological Algebra

This document was last modified on: UTC

This is a rough guide to the topics we will cover.

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 |

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I strongly encourage everyone
to use LaTex for typing their homework. If you have a mac, one
possible easy way to get started is with texshop.
If you are using linux,
there are a number of other possible ways to go, using emacs,
ghostview, etc. If you are using windows,
you're on your own, but I'm sure there's something
online. Here is a sample homework file to use: (the .tex
file, the .bib
file, and the .pdf file).
It is probably easiest to just look at the .tex file, and
start to experiment. There is also the (Not so)
short introduction to latex, which will answer most of
your questions (although typing your question into your
favorite search engine will probably work well, too).
You may also want to try http://scribtex.com/
for a cloud version.