189-570A: Higher Algebra Assignment 3

Due: Monday, November 20

1. Let k be a field and G be a finite group whose order is not divisible by the characteristic of k. Show that the group ring k[G] is isomorphic to a direct sum of rings

k[G] ≃ M_{n_{1}}(D_{1}) ⊕ ⋯ ⊕ M_{n_{r}}(D_{r}),  (1)

where each D_{i} is a (not necessarily commutative) field containing k in its center, and M_{n}(D) denotes the n × n matrix ring with entries in D.

2. For each of the following values of k and G, find the invariants r, (n_{1}, …, n_{r}) and (D_{1}, …, D_{r}) for which the isomorphism (1) holds. (For this question, you will find it useful to consult the character tables you have already constructed for various groups.)

a) G = Z/3Z, k = R.

b) G = Z/5Z, k = Q.

c) G = Z/5Z, k = R.

d) G = D_{8}, k = R.

e) G = Q, k = R.

f) G = S_{5}, k = Q.

g) G = A_{5}, k = Q.

3. Let H_{1} and H_{2} be subgroups of a finite group G. Show that the following two conditions are equivalent.

a) For all conjugacy classes C of G, the intersections C ∩ H_{1} and C ∩ H_{2} have the same cardinalty.

b) For all representations V of G, the dimensions of V^{H_{1}} and V^{H_{2}} are the same. (Here V^{H} denotes the subspace of vectors that are fixed by all elements of H.)

(Hint: You may find it useful to take another look at problem 10 of assignment 1.)

4. Let χ be a C-valued character of a finite group G. Show that |χ(g)| ≤ χ(1), and that this inequality is strict when χ is the character of a non-trivial representation, g ≠ 1, and G is a non-abelian simple group.

5. Let g be an element of order 3 in a finite group G. Show that χ(g) is rational for all characters χ, if and only if g is conjugate to g^{2}. What can you say if g has order 5?

6. Let A and B be Noetherian local rings with maximal ideals m_{A} and m_{B} respectively. Let f : A⟶B be a ring homomorphism satisfying f^{−1}(m_{B}) = m_{A}, and assume that

a) The induced homomorphism A/m_{A}⟶B/m_{B} is an isomorphism;

b) The induced homomorphism m_{A}⟶m_{B}/m_{B}^{2} is surjective;

c) B is finitely generated as an A-module, via f.

Show that f is surjective.

7. Let A be a Noetherian local ring without nilpotent elements, and let E be a finitely generated A-module. Assume that for each prime ideal p of A, the dimension of E_{p}/pE_{p} over A_{p}/pA_{p} is independent of p. Show that E is a free A-module.

8. In the case where A = Z_{p}[[T]] and E ⊂ A × A is the set of (f, g) with f(0) = g(0), show that E is not free by exhibiting two prime ideals p_{1} and p_{2} of A for which the dimensions of E_{p_{j}}/p_{j}E_{p_{j}} are different.

9. Show that if R is a Noetherian ring and S is a multiplicative subset of R then S^{−1}R is Noetherian. (Hence, in particular, the class of Noetherian rings is preserved by localisation.)