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111.16 Hilbert Nullstellensatz

Exercise 111.16.1. A silly argument using the complex numbers! Let ${\mathbf C}$ be the complex number field. Let $V$ be a vector space over ${\mathbf C}$. The spectrum of a linear operator $T : V \to V$ is the set of complex numbers $\lambda \in {\mathbf C}$ such that the operator $T - \lambda \text{id}_ V$ is not invertible.

  1. Show that $\mathbf{C}(X)$ has uncountable dimension over ${\mathbf C}$.

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

  3. Show that if a finitely generated ${\mathbf C}$-algebra $R$ is a field, then the map ${\mathbf C}\to R$ is an isomorphism.

  4. Show that any maximal ideal ${\mathfrak m}$ of ${\mathbf C}[x_1, \ldots , x_ n]$ is of the form $(x_1-\alpha _1, \ldots , x_ n-\alpha _ n)$ for some $\alpha _ i \in {\mathbf C}$.

Remark 111.16.2. Let $k$ be a field. Then for every integer $n\in {\mathbf N}$ and every maximal ideal ${\mathfrak m} \subset k[x_1, \ldots , x_ n]$ the quotient $k[x_1, \ldots , x_ n]/{\mathfrak m}$ 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 it in the case $k = {\mathbf C}$.

Exercise 111.16.3. Let $k$ be a field. Please use Remark 111.16.2.

  1. Let $R$ be a $k$-algebra. Suppose that $\dim _ k R < \infty $ and that $R$ is a domain. Show that $R$ is a field.

  2. Suppose that $R$ is a finitely generated $k$-algebra, and $f\in R$ not nilpotent. Show that there exists a maximal ideal ${\mathfrak m} \subset R$ with $f\not\in {\mathfrak m}$.

  3. Show by an example that this statement fails when $R$ is not of finite type over a field.

  4. Show that any radical ideal $I \subset {\mathbf C}[x_1, \ldots , x_ n]$ is the intersection of the maximal ideals containing it.

Remark 111.16.4. This is the Hilbert Nullstellensatz. Namely it says that the closed subsets of $\mathop{\mathrm{Spec}}(k[x_1, \ldots , x_ n])$ (which correspond to radical ideals by a previous exercise) are determined by the closed points contained in them.

Exercise 111.16.5. Let $A = {\mathbf C}[x_{11}, x_{12}, x_{21}, x_{22}, y_{11}, y_{12}, y_{21}, y_{22}]$. Let $I$ be the ideal of $A$ generated by the entries of the matrix $XY$, with

\[ X = \left( \begin{matrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix} \right) \quad \text{and}\quad Y = \left( \begin{matrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{matrix} \right). \]

Find the irreducible components of the closed subset $V(I)$ of $\mathop{\mathrm{Spec}}(A)$. (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.

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