General
The task of this course is to give an introduction to the main tools of homological algebra. This is a 6 credits course of 42 hour lectures. For the exams some homeworks & seminars on a specific topic will be assigned during the lectures. See also the Algant local page for schedules, etc.
Course (6 credits)
Description
Homological Algebra studies chain complexes and derived functors with applications to several mathematical subjects: from algebraic topology to commutative algebra and algebraic geometry, from number theory to representation theory, from operator algebras to complex and non-commutative geometry.
See the following keywords Wiki Homological Algebra, Wiki Derived Category & Wiki Local Cohomology for a sample. See also Chuck Weibel's History of Homological Algebra
Prerequisites
We assume known the basic notions from undergraduate algebra & topology. The students should be familiar with the language of categories & functors.
Notes Some notes are
here and you can find the exercises at the Ariel site Homological Algebra
Program (keywords & outline)
homotopy & homology
chain complexes & chain homotopies
projective & injective resolutions
derived functors
Tor & Ext
sheaf cohomology
spectral sequences & applications
homotopy/derived categories
Exams
Some homeworks will be assigned during the lectures. Preferably send the solutions to me as a pdf file by email. Next a seminar on your favorite subject will be assigned according to the following themes here listed below. Please check here the dates for the current academic year. You need to fill the SIFA form online here to get your name listed for the exam (but the page is in italian so you might need help). Those who for some reason wish to postpone the examination procedure can make a motivated agreement (not by email) with me for a later date if there are good reasons for that.
Themes for the Homological Algebra seminars.
Homological Dimension For example, Serre's homological characterization of regular local rings.
Local Cohomology For example, Cohen-Macaulay rings & Grothendieck local duality.
Group Cohomology For example, group extensions and universal central extension.
Galois cohomology For example, Hochschild-Serre spectral sequence & Hilbert's Theorem 90.
Sheaf Cohomology For example, Etale cohomology and/or Grothendieck-Verdier duality.
Lie Algebra Cohomology For example, Whitehead's vanishing lemmas.
Simplicial Categories For example, Dold-Kan correspondence.
Derived Categories For example, total derived functors.