Homologisk Algebra - Høst 2005

Instructor: Markus Schmidmeier
Guest Professor at Institutt for matematiske fag

Homological Algebra

Autumn semester 2005

Welcome to my course Introduction to Homological Algebra (MA 3204) We meet Wednesdays, 14:15-16:00, and Fridays, 8:15-10:00 in Kjel 22 (Varmetek., 1. etasje). For an updated version of this page please go to Homological Algebra Fall 2005.

Textbook

We use the textbook by Joseph J. Rotman, Introduction to Homological Algebra, Academic Press 1979.

Topics

Categories and Functors (chapter 1) A first introduction to categories and functors; we meet examples from several areas in mathematics but focus on representations for which we compute homomorphism groups.

Constructions on modules (chapter 2) We recall basic constructions in module theory, in particular: Tensor products, sums, products, limits, inverse limits, pushouts, and pullbacks, and describe them in terms of homomorphism groups. Many of these constructions are given explicitely for quiver representations, so besides learning about the theory of these constructions we will gain some practise in handling them.

Homological objects (chapter 3&4) We introduce modules with special homological properties: Free modules, projective modules, flat modules, and injective modules. We give explicite constructions in the case of quiver representations, but consider also modules over the integers and the polynomial ring.

Diagram chasing (chapter 6) Homology groups describe how much a chain of modules differs from an exact sequence. Diagrams help to handle several such sequences and to compute homology groups.

Ext (Chapter 7) Ext groups describe how ``larger'' objects can be built by stacking ``more basic'' ones on top of each other, very much like in lego constructions. Ext is introduced as a derived functor but we will see that Ext groups really deal with short exact sequences. In the case of quiver representations we will see how Ext groups can be computed explicitely.

Tor (Chapter 8) Tor groups are derived from tensor products and help understand flat, torsion and torsion free modules.

Homework Problems: Some homework problems will be assigned every week; under this link you can find the problem sets. Many questions on the midterm exams and the final will be related to homework problems. In addition, you can earn extra credit for nice presentations of your solutions.

Midterm exams: We are going to have three midterm exams of 45 minutes each; the best two will count for together 1/5 of the grade.

Final Exam: It the final exam will be an oral exam which counts for 4/5 of the grade. See Studiehandbok.

Contact Me

Office hours: Monday, 14:00-16:00, in Sentralbygg II, 6. etasje, rom 650, and after class

Telephone: 735-9 17 03

E-mail: schmidme@math.ntnu.no

Last modified: by Markus Schmidmeier

Categories and Functors (chapter 1)	A first introduction to categories and functors; we meet examples from several areas in mathematics but focus on representations for which we compute homomorphism groups.
Constructions on modules (chapter 2)	We recall basic constructions in module theory, in particular: Tensor products, sums, products, limits, inverse limits, pushouts, and pullbacks, and describe them in terms of homomorphism groups. Many of these constructions are given explicitely for quiver representations, so besides learning about the theory of these constructions we will gain some practise in handling them.
Homological objects (chapter 3&4)	We introduce modules with special homological properties: Free modules, projective modules, flat modules, and injective modules. We give explicite constructions in the case of quiver representations, but consider also modules over the integers and the polynomial ring.
Diagram chasing (chapter 6)	Homology groups describe how much a chain of modules differs from an exact sequence. Diagrams help to handle several such sequences and to compute homology groups.
Ext (Chapter 7)	Ext groups describe how ``larger'' objects can be built by stacking ``more basic'' ones on top of each other, very much like in lego constructions. Ext is introduced as a derived functor but we will see that Ext groups really deal with short exact sequences. In the case of quiver representations we will see how Ext groups can be computed explicitely.
Tor (Chapter 8)	Tor groups are derived from tensor products and help understand flat, torsion and torsion free modules.

Office hours:	Monday, 14:00-16:00, in Sentralbygg II, 6. etasje, rom 650, and after class
Telephone:	735-9 17 03
E-mail:	schmidme@math.ntnu.no