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.12. 
Monday
January 29 
Simplicial sets
continued Simplicial topological spaces and the EilenbergZilber 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.34. 
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 
Simplicial
sets continued Complexes. Simplicial abelian groups, Cech complex, singular chains, examples, homotopies of complexes, continued. 
Gelfand and Manin Section I.7  
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.67 
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.12 
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.34 
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 

March 2630  SPRING BREAK 
SPRING BREAK 
SPRING BREAK 
Monday
April 2 
Introduction to
derived functors 

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, quasiisomorphism. 
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.23 
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.56 

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.