Homework 1 for Group Cohomology, MATH 682

Unless specified otherwise, k is a field. Problem 0. Find as many (non-identical) explanations as you can to justify that H^{i}(G, kG) = 0 for i > 0.

Problem 1. Let P (M) be the projective cover of a G-module M and S be a simple G-module. Show that

Hom_{G}(P (M), S) = Hom_{G}(M, S).

Problem 2. Let M be a finite dimensional G module, and M^{#} be the linear dual of M, M^{#} = Hom_{k}(M, k).

(a) Show that M is a direct summand of M ⊗ M^{#} ⊗ M.

(b) Conclude that the following are equivalent:

(1) M is projective

(2) End_{k}(M, M) = M ⊗ M^{#} is projective

(3) M^{#} is projective

(4) M is injective

(c) Conclude that kG is a self-injective algebra, i.e. is an injective module over itself.

(d) Give an example of a ring R which is not an injective module over itself.

Problem 3. Let 0 → M_{1} → M_{2} → M_{3} → 0 be an exact sequence of Gmodules. Determine explicitly the connecting homomorphism in the long exact sequence induced in cohomology,

δ : H^{0}(G, M_{3}) → H^{1}(G, M_{1}).

For the next two problems, denote by I the augmentation ideal of kG, i.e. I = Ker {ϵ : kG → k} Problem 4. Show that

Ext_{G}^{1}(k, k) ≃ Hom_{k}(I/I^{2}, k).

Problem 5. Prove that H_{1}(G, ℤ) = G^{ab} using short exact sequence of Gmodules 0 → I → ℤG → ℤ → 0.

Problem 6. Compute H^{i}(ℤ/p, ℤ/p) and H_{i}(ℤ/p, ℤ/p).

Problem 7. Compute H^{i}(ℤ/3 × ℤ/2, ℤ) and H_{i}(ℤ/3 × ℤ/2, ℤ) .

Problem 8. Show that H^{1}(G, ℤ) = 0 for any finite group G.

Problem 9. (Hilbert’s theorem 90: multiplicative version). Let L/K be a finite Galois extension of fields, with Galois group G. G clearly acts on the group of units of L, L^{∗}. Hilbert’s theorem 90 states that if θ : G → L^{∗} is a derivation such that θ(gh) = gθ(h)θ(h), then θ(g) = (gx)/x.

(a) Show that Hilbert’s theorem 90 reformulates as

H^{1}(G, L^{∗}) = 0.

(b) Show that H^{2}(G, L^{∗}) does not necessarily vanish. (Hint: look, for example , at ℂ/ℝ).