Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

# Homological Algebra

Welcome to my Homological Algebra course! We meet Monday, Wednesday and Friday 2:00 - 2:50 p.m. in BU 112. The course is a combined graduate and undergraduate course (CRN: 24306 MAT 4930-005 and CRN: 22698 MAS 6396-001).

Prerequisites

Some familiarity with modules, as covered in the book by Herstein, Topics in Algebra, Chapter 4, or in the book by Ringel and Schröer, Representation Theory of Algebras, Part 1.

Topics

An introduction to homological algebra, in particular for students with interests in algebra, geometry or topology. We will be covering:

• Constructions of modules and homomorphisms: short exact sequences, push-outs, pull-backs, snake lemma.
• Functors, and modules defined by functors: Hom and tensor functors; free, flat, projective and injective modules.
• Complexes: homotopy of morphisms, long exact homology sequence, projective resolutions, Ext and Tor.
• More about Ext: long exact sequence, vector space structure of first extension group.
• Related topics, e.g. an outlook to almost split sequences.

Objectives

• Recognize the use of categories and functors in various areas in mathematics.
• Use concepts like projectivity, resolutions and Ext-groups to study systems of vector spaces and linear maps.
• Investigate complexes using the long exact homology sequence.
• Practise written and oral presentations on advanced topics in mathematics.
• Prepare a term paper using state of the art mathematical type setting software.

Textbook

C. M. Ringel and J. Schröer, Representation Theory of Algebras, 2nd preliminary version. I plan to cover much of the material in Part 3 (A-modules), Part 4 (Projective modules) and Part 5 (Homological Algebra I).

The textbook by Rotman, Homological Algebra, is a classic which is now available in the 2nd edition. I also list (in alphabetical order) several common textbooks that show the extent to which methods from homological algebra are used in algebra, geometry, topology...

• J. J. Rotman, An Introduction to Homological algebra, 2nd edition, Springer Universitext, ISBN 978-0-387-24527-0.
• F. W. Anderson, K. R. Fuller, Rings and categories of modules, Springer GTM 13 (1992).
• M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, (1997).
• A. Skowroński, K. Yamagata, Frobenius Algebras I, Basic Representation Theory, EMS Textbooks in Mathematics (2011).
• D. Eisenbud, Commutative Algebra with a view toward algebraic geometry, Springer GTM 150 (1995).
• J. R. Munkres, Elements of Algebraic Topology, Perseus Publishing (1984).

Credit

Homework:  There will be weekly homework assignments, the 10 best will count for 40% of the grade.

Presentation:   Two presentations will count for 2×20% of the grade.

Written project:   One paper, written using a professional mathematics typesetting system, will count for 20% of the grade.

Final Exam:   The class will meet on Wednesday, May 2, at 1:15 p.m.; note that we have no written final exam. Instead, final exam credit is given for a presentation and for the term paper.

Further Information