Homology in Algebra and Geometry

Carles Broto, Dolors Herbera and Wolfgang Pitsch of UAB. Santiago Zarzuela (UB)  and Victor Rotger (UPC)
Spring 2014

Homology is an ubiquitous tool which arises in a natural way when approaching a wide variety of questions from all branches of mathematics. The course will present the classical setting of homology as well as an introduction to the more recent aspects that right now are changing the subject.The aim is to stress on the applications on different areas. To emphasize this aspect some thematic days on specific subjects will be organized. These sessions will be animated by an expert of the area and will be the occasion for the interested students to show their learning of the techniques introduced during the course.

  • Introduction. The use of homology theory in different branches, through examples.
  • The classical approach: Cartan and Eilenberg.
    We introduce some categorical notions and constructions: abelian categories, tensor products (if needed), adjoint functors, limits/colimits, projective and injective objects, flat modules and the first derived functors (Tor, Ext,…)
  • The triangulated approach: Quillen’s exact categories vs. Gabriel-Zisman localization. 
    The different approaches that lead to triangulated structures will be introduced: exact categories, localization, etc. The main aim is to present the seminal theorems of Rickard on the Morita theory for riangulated categories.
  • Three thematic days:
    • A landmark for Number Theory and Algebraic Geometry: Étale cohomology.
    • A fundamental tool: Local Cohomology.
    • Further structure coming from homology theories: The Steenrod Algebra.
  • Cartan, Henri; Eilenberg, Samuel Homological algebra. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp.
  • Holm, T. and Jorgensen, P. Triangulated Categories: definitions, properties and examples. Triangulated Categories, 1–51, London Math. Soc. Lecture Note Ser., 375, Cambridge, 2010.
  • Keller, Bernhard On the construction of triangle equivalences. Derived equivalences for group rings, 155–176, Lecture Notes in Math., 1685, Springer, Berlin, 1998.
  • Weibel, Charles A. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. xiv+450 pp.
  • Neeman, Amnon Triangulated categories.  Annals of Mathematics Studies, 148. Princeton University Press, Princeton, NJ, 2001. viii+449 pp.