Lemma 29.20.11. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

1. The morphism $f$ is locally quasi-finite.

2. For every pair of 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 quasi-finite.

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 quasi-finite.

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 quasi-finite, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is locally quasi-finite 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 quasi-finite.

Proof. For a ring map $R \to A$ let us define $P(R \to A)$ to mean “$R \to A$ is quasi-finite” (see remark above lemma). We claim that $P$ is a local property of ring maps. We check conditions (a), (b) and (c) of Definition 29.14.1. In the proof of Lemma 29.15.2 we have seen that (a), (b) and (c) hold for the property of being “of finite type”. Note that, for a finite type ring map $R \to A$, the property $R \to A$ is quasi-finite at $\mathfrak q$ depends only on the local ring $A_{\mathfrak q}$ as an algebra over $R_{\mathfrak p}$ where $\mathfrak p = R \cap \mathfrak q$ (usual abuse of notation). Using these remarks (a), (b) and (c) of Definition 29.14.1 follow immediately. For example, suppose $R \to A$ is a ring map such that all of the ring maps $R \to A_{a_ i}$ are quasi-finite for $a_1, \ldots , a_ n \in A$ generating the unit ideal. We conclude that $R \to A$ is of finite type. Also, for any prime $\mathfrak q \subset A$ the local ring $A_{\mathfrak q}$ is isomorphic as an $R$-algebra to the local ring $(A_{a_ i})_{\mathfrak q_ i}$ for some $i$ and some $\mathfrak q_ i \subset A_{a_ i}$. Hence we conclude that $R \to A$ is quasi-finite at $\mathfrak q$.

We conclude that Lemma 29.14.3 applies with $P$ as in the previous paragraph. Hence it suffices to prove that $f$ is locally quasi-finite is equivalent to $f$ is locally of type $P$. Since $P(R \to A)$ is “$R \to A$ is quasi-finite” which means $R \to A$ is quasi-finite at every prime of $A$, this follows from Lemma 29.20.6. $\square$

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