Lemma 37.58.2. Let f : X \to S be a morphism of schemes which is locally of finite type. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. The following are equivalent
\mathcal{F} is of finite presentation relative to S,
for every affine opens U \subset X, V \subset S with f(U) \subset V the \mathcal{O}_ X(U)-module \mathcal{F}(U) is finitely presented relative to \mathcal{O}_ S(V).
Moreover, if this is true, then for every open subschemes U \subset X and V \subset S with f(U) \subset V the restriction \mathcal{F}|_ U is of finite presentation relative to V.
Proof.
The final statement is clear from the equivalence of (1) and (2). It is also clear that (2) implies (1). Assume (1) holds. Let S = \bigcup V_ i and f^{-1}(V_ i) = \bigcup U_{ij} be affine open coverings as in Definition 37.58.1. Let U \subset X and V \subset S be as in (2). By More on Algebra, Lemma 15.80.8 it suffices to find a standard open covering U = \bigcup U_ k of U such that \mathcal{F}(U_ k) is finitely presented relative to \mathcal{O}_ S(V). In other words, for every u \in U it suffices to find a standard affine open u \in U' \subset U such that \mathcal{F}(U') is finitely presented relative to \mathcal{O}_ S(V). Pick i such that f(u) \in V_ i and then pick j such that u \in U_{ij}. By Schemes, Lemma 26.11.5 we can find v \in V' \subset V \cap V_ i which is standard affine open in V' and V_ i. Then f^{-1}V' \cap U, resp. f^{-1}V' \cap U_{ij} are standard affine opens of U, resp. U_{ij}. Applying the lemma again we can find u \in U' \subset f^{-1}V' \cap U \cap U_{ij} which is standard affine open in both f^{-1}V' \cap U and f^{-1}V' \cap U_{ij}. Thus U' is also a standard affine open of U and U_{ij}. By More on Algebra, Lemma 15.80.4 the assumption that \mathcal{F}(U_{ij}) is finitely presented relative to \mathcal{O}_ S(V_ i) implies that \mathcal{F}(U') is finitely presented relative to \mathcal{O}_ S(V_ i). Since \mathcal{O}_ X(U') = \mathcal{O}_ X(U') \otimes _{\mathcal{O}_ S(V_ i)} \mathcal{O}_ S(V') we see from More on Algebra, Lemma 15.80.5 that \mathcal{F}(U') is finitely presented relative to \mathcal{O}_ S(V'). Applying More on Algebra, Lemma 15.80.4 again we conclude that \mathcal{F}(U') is finitely presented relative to \mathcal{O}_ S(V). This finishes the proof.
\square
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