The Stacks project

Lemma 29.16.2. Let $S$ be a scheme. Let $A$ be an Artinian local ring with residue field $\kappa $. Let $f : \mathop{\mathrm{Spec}}(A) \to S$ be a morphism of schemes. Then $f$ is of finite type if and only if the composition $\mathop{\mathrm{Spec}}(\kappa ) \to \mathop{\mathrm{Spec}}(A) \to S$ is of finite type.

Proof. Since the morphism $\mathop{\mathrm{Spec}}(\kappa ) \to \mathop{\mathrm{Spec}}(A)$ is of finite type it is clear that if $f$ is of finite type so is the composition $\mathop{\mathrm{Spec}}(\kappa ) \to S$ (see Lemma 29.15.3). For the converse, note that $\mathop{\mathrm{Spec}}(A) \to S$ maps into some affine open $U = \mathop{\mathrm{Spec}}(B)$ of $S$ as $\mathop{\mathrm{Spec}}(A)$ has only one point. To finish apply Algebra, Lemma 10.54.4 to $B \to A$. $\square$

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