Homological Algebra            

University of Massachusets, Amherst

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Current Course Notes (for Spring 2016).

These are now self contained notes on Homological Algebra. Chapters 3-9 are complete (though they could always be improved). Chapers $X$ and $Y$ are not in the final form. The missing chapters 1 and 2 are introductory:\ what is homological algebra and its origins in algebraic topology.
• Chapter 3. Categories [Complete]. Last updated on March 2.
• Chapter 4. Abelian categories [Complete]. Last updated on March 2.
• Chapter 5. Exactness properties and the derived functors [Complete].
• Chapter 6. Homotopy category of complexes [Complete]. Last updated on March 29.
• Chapters 7 and 8. Bicomplexes and Spectral sequences [Complete]. Last updated on April 25.
• Chapter 9. Derived categories [Complete]. Last updated on April 29.
• Chapter X. Sheaves. So far we have only covered the the first section (Preseaves and sheaves) and this is the only one that has been updated (March 3).
• Chapter Y. Abelian category of sheaves of modules. So far we have only covered the the first section (Preseaves and sheaves) and this is the only one that has been updated (March 3).

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Old Course Notes

• The notes will hopefully change/improve during the semester.
  • Chapters 1-3, (Wed Feb 3: we are in chapter 2.)
  • Chapters 4-9,
  • Chapters 10-13 and appendices.

• A D-modules version of the text: Chapters 0-12 + Homeworks.
  This is an older version (Spring 04?). That time the idea was to develop the theory of D-modules (algebraic theory of linear differential equations), as an example of a theory dealing with classical objects that just would not get off the ground without homological algebra.
  
  However, adding the theory of D-modules slows down the exposition. So, the only thing one finds here is this example of how Homological Algebra is used for D-modules.

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Homeworks
Homework Guidelines
• H1. Multilinear Algebra I (Feb 10).
• H2. Multilinear Algebra II and Duality over a ring (Feb 17).
• H3. Multilinear Algebra III, Categories and Sheaves (Feb 24). The last problem involves line bundles on the projective line. We have not covered this material yet (section on sheaves), so the problem can be postponed til next homework.
• H4. Multilinear and Homological Algebra, Group Cohomology (March 2).
• H5. Sheaves, Koszul complex and Group Cohomology (March 9).
• H6. Koszul complex, Sheaves and Group Cohomology (March 23).
• H7. Sheaves, Ext functors and Group Cohomology (March 23).
• H8. Bicomplexes and resolutions of complexes (March 23).
• H9. Spectral sequences. (due April 21).
(March 23).
• H10. Ext, Tor and group (co)homology (due April 21).

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Office Hours: by arrangement.
Syllabus: to appear.