Mathematics 894 Problem Set 7

Due date: Friday March 7.

1. Let k be a field, R = k[x, y] and I =< x, y > an ideal of R. Find the kernel of the R-module homomorphims I ⊗_{R}I → R that sends a ⊗ b to ab, a, b ∈ I.

2. Let R be a not necessarily commutative ring, and let F = R^{2} regarded as a left R-module. Let e_{1} = (1, 0) and e_{2} = (0, 1) be the standard basis of F.

(a) If the elements of F are represented by row vectors of length 2, show that for any linear map φ : F → F there is a unique 2 × 2 matrix A such that φ(x) = xA for any x ∈ F.

(b) We have seen that F^{∗} = Hom_{R}(F, R) is a right R-module, with fr defined by (fr)(x) = f(x)r for x ∈ F and r ∈ R. Furthermore, if we define e_{i}^{∗} ∈ F^{∗} by e_{i}^{∗}(e_{j}) = δ_{i,j}, then F^{∗} is a free right R module with basis {e_{j}^{∗}}. Explain how we can represent elements of F^{∗} by matrices, in such a way that if f ∈ F^{∗} then f(x), f ∈ F^{∗}, x ∈ F can be given by matrix multiplication.

(c) If elements of F^{∗} are represented by matrices as in (b), show explicitly how φ^{∗} : F^{∗} → F^{∗} can be given by matrix multiplication. Explain your answer.

3. Let k be a field and let V be a two dimensional vector space over k with basis e_{1}, e_{2}.

(a) Write down bases for S^{2}V and Λ^{2}V.

(b) If φ : V → V has eigenvalues λ_{1}, λ_{2} ∈ k (not necessarily distinct) find the eigenvalues of S^{2}(φ) : S^{2}V → S^{2}V and of Λ^{2}(φ) : Λ^{2}(V) → Λ^{2}(V).

4. Working over the ring ℤ, let N = ℤ/2ℤ, M = ℤ/4ℤ and i : N → M sending 1 to \overline{2}. Show that T (φ) : T (N) → T (M) is not an inclusion. What about S(φ) : S(N) → S(M)?