Sebastian Casalaina

Homework and Syllabus

Homological Algebra

MATH 6290 Spring 2018

Homework is due in class and must be stapled, with your name and homework number on it, to receive credit.

Please read the suggested texts before class, and then after class make sure to attempt the homework for the sections we covered that day.

An asterix * indicates that a homework assignment has not been finalized.


Date Topics Reading Homework
Wednesday January 17
Introduction to the course
Review the theory of rings and modules, such as in:

D. Dummit and R. Foote, Abstract Algebra (3rd Edition), Wiley, 2004, Chapter 10, or,

S. Lang, Algebra, Springer, 2002, Chapter 3, or,

P. Aluffi, Algebra: Chapter 0, Graduate Series in Mathematics, AMS, 2009, Chapter 3,

or any comparable text on the subject.

Friday January 19
Simplicial sets
Triangulated spaces.  Definition, examples, skeleton, triangulation of a product of simplices.
S. Gelfand and Y. Manin, Methods of Homological Algebra (2nd Edition), Monographs in Mathematics, Springer, 2003.  Section I.1 HW 1

Write a paragraph or two telling me about your mathematical background, and your goals for this class.

Review the theory of rings and modules.  Do a few exercises; these are not to be turned in.
Monday January 22
Simplicial sets continued
Triangulated spaces continued.
Gelfand and Manin Section I.1

Wednesday January 24
Simplicial sets continued
Simplicial sets. Definition, nerve, singular simplices, triangulated spaces, examples, skeleton, dimension.
Gelfand and Manin Section I.2

Friday January 26
Simplicial sets continued
Simplicial sets continued.
Gelfand and Manin Section I.2 HW 2*

Gelfand and Manin Exercises I.1-2.
Monday January 29
Simplicial sets continued
Simplicial topological spaces and the Eilenberg--Zilber Theorem.  Three descriptions of the product of simplices, geometric realization of a bisimplicial set.
Gelfand and Manin Section I.3
Wednesday January 31
Simplicial sets continued
Homology and cohomology.  Chains and cochains, complexes, geometry of chains, coefficients.
Gelfand and Manin Section I.4
Friday February 2
Simplicial sets continued
Sheaves.  Examples, definition, presheaves and sheaves of structured sets, germs and fibers, main classes of sheaves, sheaves of functions.
Gelfand and Manin Section I.5 HW 3*

Gelfand and Manin Exercises I.3-4.
Monday February 5
Simplicial sets continued
Sheaves continued.
Gelfand and Manin Section I.5

R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.  Chapter 2 Section 1.

Wednesday February 7
Simplicial sets continued
Sheaves continued.
Gelfand and Manin Section I.5

Friday February 9
Simplicial sets continued
Sheaves continued.
Gelfand and Manin Section I.5 HW 4*

Gelfand and Manin Exercises I.5

Hartshorne Exercises II.1
Monday February 12
Simplicial sets continued
Exact sequences.  Exact sequences, morphisms of complexes, the boundary homomorphism, coefficient systems.
Gelfand and Manin Section I.6
Wednesday February 14
Simplicial sets continued
Complexes.  Simplicial abelian groups, Cech complex, singular chains, examples, homotopies of complexes.
Gelfand and Manin Section I.7
Friday February 16
Review and possible exam
Review and possible exam
Review and possible exam
Monday February 19
Review


Wednesday February 21
Category theory
Introduction to categories.



Friday February 23
Category theory continued
Introduction to categories continued, the language of categories and functors. Definitions, examples.
Gelfand and Manin Section II.1
HW 5*

Gelfand and Manin Section Exercises I.6-7
Monday February 26
Category theory continued
Equivalence of categories.  Isomorphism, definition of equivalence, examples.
Gelfand and Manin Section II.2

Wednesday February 28
Category theory continued
Representable functors.  Definitions, examples, group objects, examples, limits, colimits, adjoints.
Gelfand and Manin Section II.3
Friday March 2
Category theory continued
Representable functors continued.
Gelfand and Manin Section II.3 HW 6*

Gelfand and Manin Section II.1-2
Monday March 5
Category theory continued
Categories in geometry and topology.  Examples, locally ringed spaces, supercommutativity, Grothendieck topologies, sites, sheaves, nerves, Hom.
Gelfand and Manin Section II.4
Wednesday March 7
Category theory continued
Categories in geometry and topology continued.
Gelfand and Manin Section II.4
Friday March 9
Category theory continued
Categories in geometry and topology continued.
Gelfand and Manin Section II.4 HW 7*

Gelfand and Manin Section II.3-4
Monday March 12
Category theory continued
Additive and abelian categories.  Definitions, kernels, cokernels, sheaves and presheaves, filtered abelian groups, topological abelian groups.
Gelfand and Manin Section II.5



Wednesday March 14
Category theory continued
Additive and abelian categories continued.
Gelfand and Manin Section II.5
Friday March 16 Category theory continued
Functors in abelian categories.  Definitions, injectivity, projectivity, divisibility, flatness, acyclic objects, inverse and direct images, adjunction.
Gelfand and Manin Section II.6 HW 8*

Gelfand and Manin Section II.5
Monday March 19
Category theory continued
Functors in abelian categories continued.
Gelfand and Manin Section II.6
Wednesday March 21
Category theory continued
Limits and colimits in categories.  Definitions and examples.
C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994, Appendix A

Friday March 23
Review and possible exam
Review and possible exam
Review and possible exam
March 26--30 SPRING BREAK
SPRING BREAK
SPRING BREAK
Monday April 2
Review


Wednesday April 4
Category theory continued
Adjoint functors.  Definitions and examples.

Weibel Appendix A, Section 1.6,  and Theorem 2.6.10.

Friday April 6
Category theory continued
Adjoint functors continued.
Weibel Appendix Z, Section 1.6, and Theorem 2.6.10.
HW 9*

Gelfand and Manin Section II.6
Monday April 9
Derived categories and derived functors
Complexes. Generators and relations, homotopies, quasi-isomorphism.



Gelfand and Manin Section III.1
Wednesday April 11
Derived categories continued
Derived categories and localization.  Definitions, splittings, localization, variants.
Gelfand and Manin Section III.2
Friday April 13 Derived categories continued
Triangles as genearlized exact triples.  Translation, cylinder, cone, definitions, long exact sequence.
Gelfand and Manin Section III.3 HW 10*

Gelfand and Manin Section III.1
Monday April 16
Derived categories continued
Derived category as the localization of the homotopy category.  Definitions, basic results, additivity of the derived category.
Gelfand and Manin Section III.4
Wednesday April 18
Derived categories continued
The structure of the derived category.  Objects as complexes, Hom, Homological dimension, examples, adjoining a variable, complexes of injectives and the derived category.
Gelfand and Manin Section III.5
Friday April 20
Derived categories continued
Derived functors.  Motivation, adapted classes of objects, construction of the derived functor, uniqueness, naturality, examples.
AGelfand and Manin Section III.6 HW 11*

Gelfand and Manin Section III.2-3
Monday April 23
Derived categories continued.
Derived functor of the composition, Grothendieck spectral sequence.  Main result, introduction to spectral sequences, filtered complexes, double complexes, hypercohomology.
Gelfand and Manin Section III.7
Wednesday April 25
Derived categories continued.
Sheaf cohomology.  Direct images and cohomology, tensor products and flat sheaves, inverse images and tensor products, higher direct images with compact support, dimension, upper shriek, dualizing complex.



Friday April 27
Further topics
HW 12*


Gelfand and Manin Section III.5-6
Monday April 30
Review


Wednesday May 2
Review

Friday May 4 NO CLASS
NO CLASS
NO CLASS
Sunday May 6
FINAL EXAM 1:30 PM - 4:00 PM HUMN 125 (Lecture Room) FINAL EXAM
FINAL EXAM

I strongly encourage everyone to use LaTeX for typing homework.  If you have a mac, one possible easy way to get started is with texshop. If you are using linux, there are a number of other possible ways to go, using emacs, ghostview, etc. If you are using windows, you're on your own, but I'm sure there's something online. Here is a sample homework file to use: (the .tex file, the .bib file, and the .pdf file).  This site can help you find LaTeX symbols by drawing: http://detexify.kirelabs.org/classify.html.  You may also want to try https://cocalc.com (formerly https://cloud.sagemath.com/) or https://www.sharelatex.com/ for a cloud version.