Homotopical Algebra 2010/11
- General
The main task is to give an introduction to the methods of homotopical algebra with a special regard to algebraic geometry. This is a 6 credits advanced course of 42 hour lectures.
For the he exams some homeworks & seminars on a specific topic will be assigned during the lectures.
See Algant local page for schedules, etc. See here for exams.
- Homotopical Algebra (6 credits)
- Description
Homotopical Algebra studies homotopies appearing in several mathematical spots: from topology to algebraic geometry and more. Notably, homotopical algebra can be regarded as a generalization of homological algebra.
- Prerequisites
We assume known the basic notions from algebraic topology & homological algebra. The students should be familiar with fundamental group, singular homology/cellular homology, CW-complexes and related basic computations. For example, see A. Hatcher Algebraic topology Cambridge Univ. Press, 2002 In a few lectures we will treat some topology using J. P. May A concise course in algebraic topology Chicago Lectures in Mathematics 1999.
For homological algebra we just assume known the basics on chain complexes as explained in the first chapter of C. Weibel's book An introduction to Homological Algebra Cambridge Univ. Press, 1994.
- Notes Course notes are not available as yet
- Program (key words & outline)
- homotopy & homology
- weak equivalences & quasi isomorphisms
- fibrations & cofibrations
- simplicial structures & geometric realisation
- model categories & homotopy categories
- Quillen functors & derived functors
- A1-homotopy & motivic homotopy theories
- References
Extras
Homotopy
- P.S. Hirschhorn: Model categories and their localizations Math Surveys & Monographs Vol. 99 AMS, 2003.
- D. Dagger: Universal homotopy theories Adv. Math. 164 (2001), no. 1, 144-176
- P. G. Goerss & J. F. Jardine: Simplicial homotopy theory Progress in Mathematics Vol. 174 Birkhauser Verlag, 1999.
- W.G. Dwyer & J. Spalinski: Homotopy theories and model categories in Handbook of Algebraic Topology I.M. James (ed.) North-Holland, 1995
- D. Quillen: "Homotopical Algebra" Springer LNM 43, 1967
- P. Gabriel & M. Zisman: "Calculus of fractions and homotopy theory" Springer, 1967
Motives
- B.I. Dundas, M. Levine, P.A. Ostvaer & V. Voevodsky:
Motivic Homotopy Theory Lectures at a Summer School in Nordfjordeid (Norway, August 2002) Springer Universitext, 2007
- F. Morel & V. Voevodsky: A1-homotopy theory of schemes Publications Mathématiques de l'IHÉS Vol. 90 (1999), p. 45-143
- C. Mazza, V. Voevodsky & C. Weibel: Lecture Notes on Motivic Cohomology Clay Mathematics Monographs, Vol. 2, 2006 (Lectures given by V. Voevodsky at IAS in 1999-2000)
K-Theory
- C. Weibel: An introduction to algebraic K-theory this is a graduate textbook in progress available on line at C. Weibel's home page
- E.M. Friedlander & D.R. Grayson (ed.) Handbook of K-theory Vol. 1, 2 Springer-Verlag, 2005
- W. Fulton & S. Lang: "Riemann-Roch Algebra" Springer Grundlehren der mathematischen Wissenschaften, Vol. 277, 1985.