Situation 38.20.1. Let $f : X \to S$ be a morphism of schemes. Let $u : \mathcal{F} \to \mathcal{G}$ be a homomorphism of quasi-coherent $\mathcal{O}_ X$-modules. For any scheme $T$ over $S$ we will denote $u_ T : \mathcal{F}_ T \to \mathcal{G}_ T$ the base change of $u$ to $T$, in other words, $u_ T$ is the pullback of $u$ via the projection morphism $X_ T = X \times _ S T \to X$. In this situation we can consider the functor

38.20.1.1
$$\label{flat-equation-iso} F_{iso} : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\} & \text{if} & u_ T \text{ is an isomorphism}, \\ \emptyset & \text{else.} & \end{matrix} \right.$$

There are variants $F_{inj}$, $F_{surj}$, $F_{zero}$ where we ask that $u_ T$ is injective, surjective, or zero.

There are also:

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