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 Ext
^{1}and extensions - 01 Popular categories
- 00 Introduction and overview

Sample phenomena illustrating homological ideas in nature:

- [ 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*

- Weibel

Unless explicitly noted otherwise, everything here, work by Paul Garrett, is licensed under a Creative Commons Attribution 3.0 Unported License. ... [

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