The Stacks project

Lemma 37.57.1. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. there exist an affine open covering $S = \bigcup V_ j$ and for each $j$ an affine open covering $f^{-1}(V_ j) = \bigcup U_{ji}$ such that $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_{ij})$ is a pseudo-coherent ring map,

  2. for every pair of affine opens $U \subset X$, $V \subset S$ such that $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is pseudo-coherent, and

  3. $f$ is locally of finite type and $\mathcal{O}_ X$ is pseudo-coherent relative to $S$.

Proof. To see the equivalence of (1) and (2) it suffices to check conditions (1)(a), (b), (c) of Morphisms, Definition 29.14.1 for the property of being a pseudo-coherent ring map. These properties follow (using localization is flat) from More on Algebra, Lemmas 15.81.12, 15.81.11, and 15.81.16.

If (1) holds, then $f$ is locally of finite type as a pseudo-coherent ring map is of finite type by definition. Moreover, (1) implies via Lemma 37.56.7 and the definitions that $\mathcal{O}_ X$ is pseudo-coherent relative to $S$. Conversely, if (3) holds, then we see that for every $U$ and $V$ as in (2) the ring $\mathcal{O}_ X(U)$ is of finite type over $\mathcal{O}_ S(V)$ and $\mathcal{O}_ X(U)$ is as a module pseudo-coherent relative to $\mathcal{O}_ S(V)$, see Lemmas 37.56.6 and 37.56.7. This is the definition of a pseudo-coherent ring map, hence (2) and (1) hold. $\square$


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