The Stacks project

Exercise 111.21.1. Let $k$ be your favorite algebraically closed field. Below $k[x]$ and $k[x, y]$ denote the polynomial rings.

1. For every integer $n \geq 0$ find a finite type extension $k[x] \subset A$ of domains such that the spectrum of $A/xA$ has exactly $n$ irreducible components.

2. Make an example of a finite type extension $k[x] \subset A$ of domains such that the spectrum of $A/(x - \alpha )A$ is nonempty and reducible for every $\alpha \in k$.

3. Make an example of a finite type extension $k[x, y] \subset A$ of domains such that the spectrum of $A/(x - \alpha , y - \beta )A$ is irreducible¹ for all $(\alpha , \beta ) \in k^2 \setminus \{ (0, 0)\} $ and the spectrum of $A/(x, y)A$ is nonempty and reducible.

Exercise 111.21.2. Let $k$ be your favorite algebraically closed field. Let $n \geq 1$. Let $k[x_1, \ldots , x_ n]$ be the polynomial ring. Set $\mathfrak m = (x_1, \ldots , x_ n)$. Let $k[x_1, \ldots , x_ n] \subset A$ be a finite type extension of domains. Set $d = \dim (A)$.

1. Show that $d - 1 \geq \dim (A/\mathfrak m A) \geq d - n$ if $A/\mathfrak mA \not= 0$.

2. Show by example that every value can occur.

3. Show by example that $\mathop{\mathrm{Spec}}(A/\mathfrak m A)$ can have irreducible components of different dimensions.