The Stacks project

Situation 99.16.3. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper, flat, and of finite presentation. We denote $\mathcal{C}\! \mathit{omplexes}_{X/B}$ the category whose objects are triples $(T, g, E)$ 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. $E$ is an object of $D(\mathcal{O}_{X_ T})$ satisfying conditions (1) and (2) of Lemma 99.16.2.

A morphism $(T, g, E) \to (T', g', E')$ 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. $\varphi : L(h')^*E' \to E$ is an isomorphism of $D(\mathcal{O}_{X_ T})$ where $h' : X_ T \to X_{T'}$ is the base change of $h$.