The Stacks project

Lemma 29.16.1. Let $S$ be a scheme. Let $k$ be a field. Let $f : \mathop{\mathrm{Spec}}(k) \to S$ be a morphism. The following are equivalent:

  1. The morphism $f$ is of finite type.

  2. The morphism $f$ is locally of finite type.

  3. There exists an affine open $U = \mathop{\mathrm{Spec}}(R)$ of $S$ such that $f$ corresponds to a finite ring map $R \to k$.

  4. There exists an affine open $U = \mathop{\mathrm{Spec}}(R)$ of $S$ such that the image of $f$ consists of a closed point $u$ in $U$ and the field extension $\kappa (u) \subset k$ is finite.

Proof. The equivalence of (1) and (2) is obvious as $\mathop{\mathrm{Spec}}(k)$ is a singleton and hence any morphism from it is quasi-compact.

Suppose $f$ is locally of finite type. Choose any affine open $\mathop{\mathrm{Spec}}(R) = U \subset S$ such that the image of $f$ is contained in $U$, and the ring map $R \to k$ is of finite type. Let $\mathfrak p \subset R$ be the kernel. Then $R/\mathfrak p \subset k$ is of finite type. By Algebra, Lemma 10.33.2 there exist a $\overline{f} \in R/\mathfrak p$ such that $(R/\mathfrak p)_{\overline{f}}$ is a field and $(R/\mathfrak p)_{\overline{f}} \to k$ is a finite field extension. If $f \in R$ is a lift of $\overline{f}$, then we see that $k$ is a finite $R_ f$-module. Thus (2) $\Rightarrow $ (3).

Suppose that $\mathop{\mathrm{Spec}}(R) = U \subset S$ is an affine open such that $f$ corresponds to a finite ring map $R \to k$. Then $f$ is locally of finite type by Lemma 29.15.2. Thus (3) $\Rightarrow $ (2).

Suppose $R \to k$ is finite. The image of $R \to k$ is a field over which $k$ is finite by Algebra, Lemma 10.35.18. Hence the kernel of $R \to k$ is a maximal ideal. Thus (3) $\Rightarrow $ (4).

The implication (4) $\Rightarrow $ (3) is immediate. $\square$


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