Definition 37.55.1. 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 $\mathcal{O}_ S(V_ i)$.

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