EXERCISES FOR TOPOLOGY III, WS05/06

Sheet 11, January 12 2006

Solutions due on Thursday, 19th of January.

Exercise 11.1. Find an example of a connected, commutative and associative Hopf Algebra A over ℤ with two generators, such that A is not isomorphic as an algebra to any tensor product B ⊗_{ℤ} C where B and C are connected algebras on one generator.

Exercise 11.2. Repeat the previous exercise, replacing the ground ring ℤ by any non-perfect field F.

Exercise 11.3. Show that if A is a connected Hopf-algebra of finite type then P (A^{#}) = Q(A)^{#} and Q(A^{#}) = P (A)^{#}.

Exercise 11.4. Compute P (A) → Q(A) for the following values of A.

(a) A is a polynomial or exterior Hopf-Algebra on one generator in positive degree,

(b) A = P (x)/(x^{p^{k}}) over a field of characteristic p > 0, and |x| > 0,

(c) A = Γ(x), the divided power algebra over the ground ring ℤ, |x| > 0 even, with coproduct as in Exercise 5.2.

Exercise 11.5. Prove the following Lemma from the lecture : let A ⊂ B be connected associative Hopf algebras of finite type over a field F, g : B → A be an A-linear retraction of A ⊂ B, and π : B → C = F ⊗_{A} B be the projection. Then the map

ĝ : B ⟶^{Δ} B ⊗ B ⟶^{g⊗π} A ⊗ C

is an isomorphism of A-modules.

Exercise 11.6. Let G be a group and k be a field. Show that the diagonal map G → G×G, g ↦ (g, g) induces a coproduct on the group-ring kG, and that kG is an ungraded Hopf-algebra. Determine its properties (is it (co-)unital, (co-)associative, (co-)commutative ?). Show that if H ◁ G, then kH ◁ kH. Compare k(G/H) to kG//kH.

Typeset by AMS-TEX