Commutative Algebra

Excercises 4

1. A silly argument using the complex numbers! Let ℂ be the complex number field. Let V be a vector space over ℂ. The spectrum of a linear operator T : V → V is the set of complex numbers λ ∈ ℂ such that the operator T − λid_{V} is not invertible.

(a) Show that ℂ(X) = f.f.(ℂ[X]) has uncountable dimension over ℂ.

(b) Show that any linear operator on V has a nonempty spectrum if the dimension of V is finite or countable.

(c) Show that if a finitely generated ℂ-algebra R is a field, then the map ℂ → R is an isomorphism.

(d) Show that any maximal ideal 𝔪 of ℂ[x_{1}, …, x_{n}] is of the form (x_{1} − α_{1}, …, x_{n} − α_{n}) for some α_{i} ∈ ℂ.

Remark. Let k be a field. Then (†) for every integer n ∈ ℕ and every maximal ideal 𝔪 ⊂ k[x_{1}, …, x_{n}] the quotient

k[x_{1}, …, x_{n}]/𝔪 is a finite field extension of k. This will be shown later in the course. Of course (please check this) it implies a similar statement for maximal ideals of finitely generated k-algebras. The exercise above proves (†) in the case k = ℂ.

2. Let k be a field. Please use (†) in (b) below.

(a) Let R be a k-algebra. Suppose that dim_{k} R < ∞ and that R is a domain. Show that R is a field.

(b) Suppose that R is a finitely generated k-algebra, and f ∈ R not nilpotent. Show that there exists a maximal ideal 𝔪 ⊂ R with f ∉ 𝔪.

(c) Show by an example that this statement fails when R is not of finite type over a field.

(d) Show that any radical ideal I ⊂ ℂ[x_{1}, …, x_{n}] is the intersection of the maximal ideals containing it.

Remark. This is the Hilbert Nullstellensatz. Namely it says that the closed subsets of Spec k[x_{1}, …, x_{n}] (which correspond to radical ideals by a previous exercise) are determined by the closed points contained in them.

3. Let A = ℂ[x_{11}, x_{12}, x_{21}, x_{22}, y_{11}, y_{12}, y_{21}, y_{22}]. Let I be the ideal of A generated by the entries of the matrix XY, with

X = ( \begin{matrix}x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix} ) and Y = ( \begin{matrix}y_{11} & y_{12} \\ y_{21} & y_{22} \end{matrix} ) .

Find the irreducible components of the closed subset V (I) of Spec A. (I mean describe them and give equations for each of them. You do not have to prove that the equations you write down define prime ideals.) Hints:

(1) You may use the Hilbert Nullstellensatz, and it suffices to find irreducible locally closed subsets which cover the set of closed points of V (I).

(2) There are two easy components.

(3) An image of an irreducible set under a continuous map is irreducible.