## 29.14 Morphisms of finite type

Recall that a ring map $R \to A$ is said to be of finite type if $A$ is isomorphic to a quotient of $R[x_1, \ldots , x_ n]$ as an $R$-algebra, see Algebra, Definition 10.6.1.

Definition 29.14.1. Let $f : X \to S$ be a morphism of schemes.

1. We say that $f$ is of finite type at $x \in X$ if there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and an affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is of finite type.

2. We say that $f$ is locally of finite type if it is of finite type at every point of $X$.

3. We say that $f$ is of finite type if it is locally of finite type and quasi-compact.

Lemma 29.14.2. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

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

2. For all affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is of finite type.

3. There exists an open covering $S = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$, $j\in J, i\in I_ j$ is locally of finite type.

4. There exists an affine open covering $S = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that the ring map $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_ i)$ is of finite type, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is locally of finite type then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_ U : U \to V$ is locally of finite type.

Proof. This follows from Lemma 29.13.3 if we show that the property “$R \to A$ is of finite type” is local. We check conditions (a), (b) and (c) of Definition 29.13.1. By Algebra, Lemma 10.13.2 being of finite type is stable under base change and hence we conclude (a) holds. By the same lemma being of finite type is stable under composition and trivially for any ring $R$ the ring map $R \to R_ f$ is of finite type. We conclude (b) holds. Finally, property (c) is true according to Algebra, Lemma 10.22.3. $\square$

Lemma 29.14.3. The composition of two morphisms which are locally of finite type is locally of finite type. The same is true for morphisms of finite type.

Proof. In the proof of Lemma 29.14.2 we saw that being of finite type is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 29.13.5 combined with the fact that being of finite type is a property of ring maps that is stable under composition, see Algebra, Lemma 10.6.2. By the above and the fact that compositions of quasi-compact morphisms are quasi-compact, see Schemes, Lemma 26.19.4 we see that the composition of morphisms of finite type is of finite type. $\square$

Lemma 29.14.4. The base change of a morphism which is locally of finite type is locally of finite type. The same is true for morphisms of finite type.

Proof. In the proof of Lemma 29.14.2 we saw that being of finite type is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 29.13.6 combined with the fact that being of finite type is a property of ring maps that is stable under base change, see Algebra, Lemma 10.13.2. By the above and the fact that a base change of a quasi-compact morphism is quasi-compact, see Schemes, Lemma 26.19.3 we see that the base change of a morphism of finite type is a morphism of finite type. $\square$

Lemma 29.14.5. A closed immersion is of finite type. An immersion is locally of finite type.

Proof. This is true because an open immersion is a local isomorphism, and a closed immersion is obviously of finite type. $\square$

Lemma 29.14.6. Let $f : X \to S$ be a morphism. If $S$ is (locally) Noetherian and $f$ (locally) of finite type then $X$ is (locally) Noetherian.

Proof. This follows immediately from the fact that a ring of finite type over a Noetherian ring is Noetherian, see Algebra, Lemma 10.30.1. (Also: use the fact that the source of a quasi-compact morphism with quasi-compact target is quasi-compact.) $\square$

Lemma 29.14.7. Let $f : X \to S$ be locally of finite type with $S$ locally Noetherian. Then $f$ is quasi-separated.

Proof. In fact, it is true that $X$ is quasi-separated, see Properties, Lemma 28.5.4 and Lemma 29.14.6 above. Then apply Schemes, Lemma 26.21.13 to conclude that $f$ is quasi-separated. $\square$

Lemma 29.14.8. Let $X \to Y$ be a morphism of schemes over a base scheme $S$. If $X$ is locally of finite type over $S$, then $X \to Y$ is locally of finite type.

Proof. Via Lemma 29.14.2 this translates into the following algebra fact: Given ring maps $A \to B \to C$ such that $A \to C$ is of finite type, then $B \to C$ is of finite type. (See Algebra, Lemma 10.6.2). $\square$

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