Exercises in homological algebra

In the Spring of 2012 I taught a graduate course in homological algebra, following the text of Weibel: An Introduction to Homological Algebra.

In most mathematics courses, and in homological algebra especially, it is critical that one work out exercises to really internalize the material. To support this, we dedicated one day each week to go over homework during class time. Students volunteered to present their solutions to the class. As they presented, I tried to ask clarifying questions, and connect their work to the lecture material.

Because we had such a strong emphasis on homework, the lectures were somewhat fast-paced, usually covering 1.5 or 2 sections of the text each day. On the course calendar you can see a list of the topics and relevant sections we covered each day.

We also introduced spectral sequences early enough that we could have time to work with them as a black box, then go carefully through a construction (from a filtered chain complex), and then return to some applications. During this part of the class, I produced an illustrated guide to the construction of a spectral sequence!

Below, you will find the course announcement and homework exercises for the course.

What is homological algebra?

Homological algebra is a place to get answers. All kinds of subtle and interesting answers! Homological algebra is a collection of tools and techniques which are used in any field with algebra in its name: Algebra, algebraic topology, algebraic geometry, algebraic number theory, etc.

With homological algebra, we can reduce difficult questions about complex objects to basic linear algebra problems. Albeit an infinite sequence of problems, but basic problems nonetheless! One might compare this with the way an analytic function can be understood entirely by its Taylor coefficients: Derivatives are easy, and you can understand something subtle (an arbitrary function) by doing an infinite sequence of easy computations.

In practice, this means that the sheer magnitude of objects in homological algebra can be overwhelming to the novice. In this course we'll give an organized introduction and overview of the main ideas. We'll work through them in some of the classic—and most useful—applications, and we'll introduce enough special topics to pique the interest of students from a variety of backgrounds.

Outline

Our text will be Weibel's An Introduction to Homological Algebra, and most of the course will follow this text. We'll cover the basic concepts of homological algebra with most of our attention focused on central applications. At relevant points in the course, we'll foray into related topics which are of interest to the students.

Where there are expositional choices to be made, I will probably tend toward topological and categorical descriptions as a conceptual framework for the techniques of homological algebra. Background in these areas may be helpful, but is not required. The only true prerequisite for the course is familiarity with abelian groups and quotients thereof. More general familiarity with modules over commutative rings, homomorphisms, and tensor products will also be useful.

A potential syllabus is given below, although the pacing and selection of additional topics will be revised depending on the audience.

The concepts

• Chain complexes and homology
• Derived functors and derived categories
• Spectral sequences

The applications

• Homology and cohomology of spaces and of finite groups
• Ext and Tor
• The Serre spectral sequence and spectral sequences arising from exact couples

• Homological dimension
• Lie algebra co/homology
• Hochschild co/homology
• Sheaf cohomology
• Model categories and derived categories

Exercises

Here are the exercises we covered, in reverse chronological order. Also see the course calendar. The numbered exercises are taken from Weibel's text and the written descriptions are only an approximation.

Monday, 4/23:

• Exercise 7.2.2: Alernative description of g-modules in terms of Lie algebra homomorphism from g to Lie endomorphism algebra.
• Exercise 7.2.4: The invariants and coinvariants functors are, respectively, right and left adjoint to the trivial module functor.
• Corollary 7.2.5: Use the resolution given in proposition 7.2.4 to compute the co/homology of the free Lie algebra on a set X.
• Exercise 7.3.3: Show that the universal enveloping algebra functor is left adjoint to the Lie functor.
• Finish Exercise 7.3.7 showing that the universal enveloping algebra is a Hopf algebra.

Monday, 4/9:

• Exercise 5.3.2: Homology of complex projective space.
• Let K(Z,n) be a space whose nth homotopy group is the integers, and whose other homotopy groups are trivial. Note that the loop space of K(Z,n) is K(Z,n-1). Also note that K(Z,1) is the circle, S^1. Use the Serre spectral sequence and the path-loop fibration to calculate the integral homology of K(Z,n) for n = 2, 3.
• Use the Lyndon-Hochschild-Serre spectral sequence to compute the homology of Z/n from the short exact sequence
 0 --> nZ --> Z --> Z/n --> 0 
• Let n be an odd number. Follow Example 6.8.5 to calculate the cohomology of the dihedral group with 2n elements over the integers.

Monday, 4/2:

• Exercise 6.2.2: Co/homology of a product of groups.
• Exercise 6.2.4: Homology of a free group on two letters.
• Exercise 5.3.2: Homology of complex projective space.
• Let K(Z,n) be a space whose nth homotopy group is the integers, and whose other homotopy groups are trivial. Note that the loop space of K(Z,n) is K(Z,n-1). Also note that K(Z,1) is the circle, S^1. Use the Serre spectral sequence and the path-loop fibration to calculate the integral homology of K(Z,n) for n = 2, 3.
• Use the Lyndon-Hochschild-Serre spectral sequence to compute the homology of Z/n from the short exact sequence
 0 --> nZ --> Z --> Z/n --> 0 

Monday, 3/12:

• Exercise 4.5.1(2): Show that Koszul homology has an external product.
• Exercise 4.5.6: Let R be a regular local ring (so its maximal ideal is generated by a regular sequence, by Exercise 4.4.2) and let k be the residue field. Compute Ext_R^*(k, k).
• Exercise 6.1.5: Low-degree group cohomology with trivial coefficients.
• Exercise 6.1.8: The cross product for group cohomology and the Künneth sequence.

Monday, 2/27:

• Prove the porism of the Universal Coefficient Theorem from class: The theorem holds for any projective complex if the ground ring has homological dimension at most 1.
• Show that Tor and Ext are adjoint functors (2 people): define the unit and counit of the adjunction, and show that they satisfy the triangle identities.
• Exercise 4.1.2(1): Relate the projective dimension of the middle term in a short exact sequence to the projective dimensions of the ends.
• Exercise 4.1.3: If S is an R-algebra and P is a projective S-module, then the projective dimension of P over R is less than or equal to the projective dimension of S over R.

Monday, 2/20:

• Show that left adjoints commute with colimits (Theorem 2.6.10).
• Calculate Ext_{Z/p^2} (Z/p, Z/p) (see exercise 3.3.2).
• Explain Porism 3.4.2; in particular, say what a porism is.
• Followup by outlining the argument that Ext^1 classifies extensions (see theorem 3.4.3).

Wednesday, 2/8:

• Exercise 2.6.4: Review the definition of colimit and show that colimits are left adjoint to diagonal functors.
• Show that abelianization is left adjoint to inclusion from abelian groups to groups.
• Exercise 3.1.2: Tor is torsion.
• Exercise 3.2.1: Flat modules have no Tor.
• Exercise 3.3.1: Ext^1_Z (Z[1/p] , Z).

Monday, 1/30:

• Calculate Ext_Z(Z/p, Z/q) for distinct primes p and q.
• Show that Hom(M,—) is left-exact for any M.
• Show that a module M is projective if and only if Hom(M,—) is an exact functor.
• Prove Cor 2.3.2 directly: A module is injective over the integers if and only if it is divisible. Verify that Q, Q/Z, and Z/p^infty are all injective over Z.
• Find an injective resolution of Z/p and use it to calculate Ext_Z(Z/p, Z/p).

Monday, 1/23:

• 1.5.1: CA deformation retracts to 0.
• 1.5.2: Show that a chain map f is null-homotopic if and only if it extends to the cone on its domain.
• 1.5.8: Let f: A —> B be a map of chain complexes. Show that Cone(B —> Cf) is chain homotopy equivalent to the suspension of A.
• Prove that a module is projective if and only if it is a direct summand of a free module.

Friday, 1/13:

• 1.1.2: A map of chain complexes induces a map in homology; H_* is a functor.
• 1.1.4: If C is a chain complex and A is an R-module, then Hom(A,C) is a chain complex.
• 1.1.6: Homology of a graph.
• 1.1.7: Homology of the tetrahedron.
• 1.2.3: Kernels and cokernels of are kernels and cokernels.
• Verify at least two additional pieces of the Snake Lemma.
• 1.4.3: Split exact complexes.
• 1.4.5: Do at least two of the four parts.