Introduction to categories and homological algebra, spring 2013
Introduction to Categories and Homological Algebra, spring 2013
Lecturer
Gregory Arone
Prof. Arone is visiting the Department of Mathematics and Statistics as a Fulbright Scholar.
Scope
10 cu.
Type
Advanced studies
Course description
We will begin with introducing the basic notions of category theory:
categories, functors, natural transformations, direct and inverse limits, and so forth. These are very general concepts that include many familiar constructions in mathematics as special cases.
The main part of the course will be an introduction to homological algebra. We will review, as needed, basic notions about
rings and modules, tensor products and exact sequences. We will talk
about Abelian categories, chain complexes, exact and derived functors.
We will define Tor and Ext groups in a general Abelian category and
see that they encode deep information about the category. We will
see how classical notions of (co)-homology of spaces, groups, rings fit into the general framework provided by homological algebra.
In the last part of the course I hope to go over some classic applications
of the theory, or a topic of a more advanced nature. The choice of topic
will depend on our pace and on your interest. One possibility is an
introduction to Sheaf Theory, maybe culminating in a discussion of
the Riemann-Roch theorem. Or we may discuss some applications of
homological algebra to ring theory. Or to topology.
Textbooks
Main text: Joseph Rotman, "An Introduction to Homological Algebra", second edition, Springer
- (main text) Joseph Rotman, "An Introduction to Homological Algebra", second edition, Springer
- Charles Weibel, "An Introduction to Homological Algebra", Cambridge University Press
Prerequisites
The formal requirement is a basic course in Abstract Algebra (a basic
course in algebraic topology also should be adequate preparation). Before
the start of the class, I would like you to be comfortable with concepts
like Abelian group, homomorphism, kernel and quotient. Some
familiarity with rings and modules, as well as topological spaces and
continuous functions, is desirable. Any additional background in algebra
or topology is welcome, but is not necessary. The main requirement
is interest in mathematics and willingness to learn.
Lectures
Weeks 3-9 and 11-18, Monday 12-14 in room D123 and Wednesday 14-16 in room B322.
Easter holiday 25.3.-3.4.
Homeworks
(due Wednesday, Jan 23)
(due Wednesday, Jan 30)
(due Wednesday, Feb 6)
(due Wednesday, Feb 13)
(due Wednesday, Feb 20)
(due Wednesday, Feb 27)
(due Wednesday, Mar 13)
(due Wednesday, Mar 21)
(due Wednesday, April 10)
(due Wednesday, April 17)
(due Wednesday, April 24)
Exams
Bibliography
Registration
Did you forget to register? What to do.