Situation 99.5.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation. We denote $\mathcal{C}\! \mathit{oh}_{X/B}$ the category whose objects are triples $(T, g, \mathcal{F})$ where

1. $T$ is a scheme over $S$,

2. $g : T \to B$ is a morphism over $S$, and setting $X_ T = T \times _{g, B} X$

3. $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_{X_ T}$-module of finite presentation, flat over $T$, with support proper over $T$.

A morphism $(T, g, \mathcal{F}) \to (T', g', \mathcal{F}')$ is given by a pair $(h, \varphi )$ where

1. $h : T \to T'$ is a morphism of schemes over $B$ (i.e., $g' \circ h = g$), and

2. $\varphi : (h')^*\mathcal{F}' \to \mathcal{F}$ is an isomorphism of $\mathcal{O}_{X_ T}$-modules where $h' : X_ T \to X_{T'}$ is the base change of $h$.

Comment #890 by Matthew Emerton on

In condition (3), should proper over $B$ instead be proper over $T$?

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).