Homework 3 for Group Cohomology, MATH 682 due March 8

Problem 1. ( “ 5 term exact sequence” )

(a). Let E_{pq}^{2} ⇒ H_{p+q}(C) be a homology spectral sequence of a double complex C concentrated in the first quadrant. show that there is a 5 term exact sequence

H_{2}(C) → E_{20}^{2} →^{d} E_{01}^{2} → H_{1}(C) → E_{10}^{2} → 0.

(b). Write down and prove the corresponding exact sequence for cohomology.

Problem 2. Prove Universal Coefficient Theorem for cohomology: if R is a PID, C_{∗} is a chain complex which is dimension-wise flat and M is an R-module, then there is a short exact sequence

0 → Ext_{R}^{1}(H_{n−1}(C), M) → H^{n}(Hom_{R}(C, M)) → Hom_{R}(H_{n}(C), M) → 0.

In fact, this sequence is non-canonically split, but you don’t have to prove that!

Problem 3. Construct the “base-change” spectral sequence for Tor :

Let f : R → S be a ring map, M be an S module, and N be an R-module. Then there is a first quadrant homology spectral sequence

E_{pq}^{2} = Tor_{p}^{S}(Tor_{q}^{R}(M, S), N) ⇒ Tor_{p+q}^{R}(M, N),

where M is viewed as an R module via the pull-back via f.

Problem 4. Compute cohomology of S_{3}, the group of permutations on 3 elements.

Problem 5. Let D_{m} denote the dihedral group, which is the group of symmetries of the regular m − gon.

(a). Show that D_{m} = C_{m} ⋊ C_{2} where the action of C_{2} on C_{m} is via σ(a) = a^{−1} for a ∈ C_{m}, and σ being a generator of C_{2}. Here, C_{m} denote a cyclic group of order m which I don’t dare writing as ℤ/m now:).

(b). Since C_{m} is a normal subgroup in D_{m}, the factor group C_{2} acts on cohomology H^{∗}(C_{m}, ℤ). Let t be a generator of C_{m}, σ be a generator of C_{2} and g = (1, σ) ∈ D_{m}. Show that gtg^{−1} = t^{−1}. Then show that the map induced by congujation by g on H^{2n}(C_{m}, ℤ) is multiplication by (−1)^{n}.

(c). Compute cohomology of D_{m} using LHS spectral sequence.

Problem 6. Let Ch_{≥0}(Ab) be the category of complexes of abelian groups concentrated in non-negative degrees. Show that the left hyper derived functors 𝕃_{i}F restricted to Ch_{≥0}(Ab) are the left derived functors of the right exact functor H_{0}F (where F is a right exact functor on Ab itself).

Problem 7. Let R be a ring, A_{∗}, B_{∗} be bounded below chain complexes of R-modules.

(a). Show that there exist bounded below complexes P_{∗}, Q_{∗} such that

- P_{n}, Q_{n} are projective for all n for which they are non-trivial,

- P_{∗}, Q_{∗} are quasi-isomorphic to A_{∗}, B_{∗} (respectively). (Hint: consider the total complex of the Cartan-Eilenberg resolution.)

(b). If P_{∗}, Q_{∗} are as in (a), show that there are isomorphisms H_{∗}(A ⊗ B) ≃ H_{∗}(A ⊗ Q) ≃ H_{∗}(P ⊗ B) = H_{∗}(P ⊗ Q).