Commutative Algebra

Excercises 4

Let ϕ : A → B be a homomoprhism of rings. We say that the going-up theorem holds for ϕ if the following condition is satisfied:

(GU) for any 𝔭, 𝔭′ ∈ Spec(A) such that 𝔭 ⊂ 𝔭′, and for any P ∈ Spec(B) lying over 𝔭,

there exists P′ ∈ Spec(B) lying over 𝔭′ such that P ⊂ P′. Similarly, we say that the going-down theorem holds for ϕ if the following condition is satisfied:

(GD) for any 𝔭, 𝔭′ ∈ Spec(A) such that 𝔭 ⊂ 𝔭′, and for any P′ ∈ Spec(B) lying over 𝔭′, there exists P ∈ Spec(B) lying over 𝔭 such that P ⊂ P′.

1. In each of the following cases determine whether (GU), (GD) holds, and explain why. (Use any Prop/Thm/Lemma you can find, but check the hypotheses in each case.)

(a) k is a field, A = k, B = k[x].

(b) k is a field, A = k[x], B = k[x, y].

(c) A = ℤ, B = ℤ[1/11].

(d) k is an algebraically closed field, A = k[x, y], B = k[x, y, z]/(x^{2} − y, z^{2} − x).

(e) A = ℤ, B = ℤ[i, 1/(2 + i)].

(f) A = ℤ, B = ℤ[i, 1/(14 + 7i)].

(g) k is an algebraically closed field, A = k[x], B = k[x, y, 1/(xy − 1)]/(y^{2} − y).

2. Let k be an algebraically closed field. Compute the image in Spec(k[x, y]) of the following maps:

(a) Spec(k[x, yx^{−1}]) → Spec(k[x, y]), where k[x, y] ⊂ k[x, yx^{−1}] ⊂ k[x, y, x^{−1}]. (Hint: To avoid confusion, give the element yx^{−1} another name.)

(b) Spec(k[x, y, a, b]/(ax − by − 1)) → Spec(k[x, y]).

(c) Spec(k[t, 1/(t − 1)]) → Spec(k[x, y]), induced by x ↦ t^{2}, and y ↦ t^{3}.

(d) k = ℂ (complex numbers), Spec(k[s, t]/(s^{3} + t^{3} − 1)) → Spec(k[x, y]), where x ↦ s^{2}, y ↦ t^{2}.

Remark. Finding the image as above usually is done by using elimination theory.

3. Let k be a field. Show that the following pairs of k-algebras are not isomorphic:

(a) k[x_{1}, …, x_{n}] and k[x_{1}, …, x_{n+1}] for any n ≥ 1.

(b) k[a, b, c, d, e, f]/(ab + cd + ef) and k[x_{1}, …, x_{n}] for n = 5.

(c) k[a, b, c, d, e, f]/(ab + cd + ef) and k[x_{1}, …, x_{n}] for n = 6.

Remark. Of course the idea of this exercise is to find a simple argument in each case rather than applying a “big” theorem. Nonetheless it is good to be guided by general principles.