Tentative outline/list of topics

Notes, examples, supplements (Reverse chronological order. Labels will become links when targets are ready.)

• DeRham, Dolbeault theorems ... Chern classes
• Cech sheaf cohomology and connection to algebraic topology
• Presheaves and sheaves, acyclic resolutions, derived-functor sheaf cohomology
• Universal delta functors
• Derived functors, projectives, injectives, Baer's criterion
  • exercises/examples: Ext, Tor, group (co-) homology, Lie algebra (co-) homology, local cohomology
• 03 adjoints, naturality, exactness, small Yoneda lemma
  • 03x exercises/examples: standard adjoint pairs... Hom and tensor, fixed and co-fixed vectors, isotypes and co-isotypes... adjoints to forgetful functors are free objects
• 02 Yoneda's Ext¹ and extensions
• 01 Popular categories
• 00 Introduction and overview
  • 00a naive set theory
  • 00x exercises/examples
  • 00y exercises/examples

• [ Snake lemma, extensions, Gamma function ] ... [ updated Tue, 14 Jun '11, 04:46 PM]... ... Illustration of extension and uniqueness of distributions by simple homological ideas. Easiest example: homogeneous distributions and Gamma.

• In 2009-2010: MWF 12:20-1:10, Vincent 2 [Room change!]
• Prerequisites: abstract algebra ... and willingness to see examples and illustrations from all parts of mathematics
• Office hours: MWF after class, email anytime
• To integrate some things whose inter-relations are sometimes neglected, I will not follow any particular text. Of the sources listed below, Weibel might be the most relevant, and Eisenbud, both of these in terms of over-all content rather than logical ordering.
• Some standard sources:
  • Weibel Homological Algebra
  • Gelfand and Manin Methods of Homological Algebra
  • Kashiwara and Shapira Categories and Sheaves
  • Hilton and Stambach, Homological algebra
  • Cartan and Eilenberg, Homological algebra
  • Grothendieck, Tohoku J. paper
  • MacLane, Homology
  • Eisenbud Commutative Algebra with a View Toward Algebraic Geometry
  • Bourbaki Commutative Algebra
  • Atiyah and MacDonald Commutative Algebra
  • Zariski and Samuel, Commutative algebra, two volumes
  • Milnor Introduction to algebraic K-theory
  • Rosenberg Introduction to algebraic K-theory
  • Srinivas Algebraic K-theory
  • Mitchell Theory of categories
  • Freyd Abelian categories
  • MacLane Categories for the working mathematician