Niles Johnson

This is the course homepage and syllabus for Math 8030 section 52-333: Homological Algebra. Class meets

MWF, 11:15—12:05, in Boyd 328.

Class will meet following the registrar's Academic Calendar.

Office Hours: Tuesdays at 3pm and Thursdays at 2pm.

We now have some Notes on the construction of spectral sequences.

The numbered exercises are taken from Weibel's text; you should work on the version stated there rather than the approximation written here.

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.