Definition 37.58.1. 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. We say $\mathcal{F}$ is finitely presented relative to $S$ or of finite presentation relative to $S$ if there exists an affine open covering $S = \bigcup V_ i$ and for every $i$ an affine open covering $f^{-1}(V_ i) = \bigcup _ j U_{ij}$ such that $\mathcal{F}(U_{ij})$ is a $\mathcal{O}_ X(U_{ij})$-module of finite presentation relative to $\mathcal{O}_ S(V_ i)$.
37.58 Relative finite presentation
Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module. In More on Algebra, Section 15.80 we defined what it means for $M$ to be finitely presented relative to $R$. We also proved this notion has good localization properties and glues. Hence we can define the corresponding global notion as follows.
Note that this implies that $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module. If $X \to S$ is just locally of finite type, then $\mathcal{F}$ may be of finite presentation relative to $S$, without $X \to S$ being locally of finite presentation. We will see that $X \to S$ is locally of finite presentation if and only if $\mathcal{O}_ X$ is of finite presentation relative to $S$.
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. $\square$
Lemma 37.58.3. 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.
If $f$ is locally of finite presentation, then $\mathcal{F}$ is of finite presentation relative to $S$ if and only if $\mathcal{F}$ is of finite presentation.
The morphism $f$ is locally of finite presentation if and only if $\mathcal{O}_ X$ is of finite presentation relative to $S$.
Proof. Follows immediately from the definitions, see discussion following More on Algebra, Definition 15.80.2. $\square$
Lemma 37.58.4. Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is of finite presentation relative to $S$ if and only if $\pi _*\mathcal{F}$ is of finite presentation relative to $S$.
Proof. Translation of the result of More on Algebra, Lemma 15.80.3 into the language of schemes. $\square$
Lemma 37.58.5. 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. Let $S' \to S$ be a morphism of schemes, set $X' = X \times _ S S'$ and denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X'$. If $\mathcal{F}$ is of finite presentation relative to $S$, then $\mathcal{F}'$ is of finite presentation relative to $S'$.
Proof. Translation of the result of More on Algebra, Lemma 15.80.5 into the language of schemes. $\square$
Lemma 37.58.6. Let $X \to Y \to S$ be morphisms of schemes which are locally of finite type. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. If $f : X \to Y$ is locally of finite presentation and $\mathcal{G}$ of finite presentation relative to $S$, then $f^*\mathcal{G}$ is of finite presentation relative to $S$.
Proof. Translation of the result of More on Algebra, Lemma 15.80.6 into the language of schemes. $\square$
Lemma 37.58.7. Let $X \to Y \to S$ be morphisms of schemes which are locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $Y \to S$ is locally of finite presentation and $\mathcal{F}$ is of finite presentation relative to $Y$, then $\mathcal{F}$ is of finite presentation relative to $S$.
Proof. Translation of the result of More on Algebra, Lemma 15.80.7 into the language of schemes. $\square$
Lemma 37.58.8. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ be a short exact sequence of quasi-coherent $\mathcal{O}_ X$-modules.
If $\mathcal{F}', \mathcal{F}''$ are finitely presented relative to $S$, then so is $\mathcal{F}$.
If $\mathcal{F}'$ is a finite type $\mathcal{O}_ X$-module and $\mathcal{F}$ is finitely presented relative to $S$, then $\mathcal{F}''$ is finitely presented relative to $S$.
Proof. Translation of the result of More on Algebra, Lemma 15.80.9 into the language of schemes. $\square$
Lemma 37.58.9. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}, \mathcal{F}'$ be quasi-coherent $\mathcal{O}_ X$-modules. If $\mathcal{F} \oplus \mathcal{F}'$ is finitely presented relative to $S$, then so are $\mathcal{F}$ and $\mathcal{F}'$.
Proof. Translation of the result of More on Algebra, Lemma 15.80.10 into the language of schemes. $\square$
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