The Stacks project

Situation 99.14.1. We define a category $\mathcal{P}\! \mathit{olarized}$ as follows. Objects are pairs $(X \to S, \mathcal{L})$ where

  1. $X \to S$ is a morphism of schemes which is proper, flat, and of finite presentation, and

  2. $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module which is relatively ample on $X/S$ (Morphisms, Definition 29.37.1).

A morphism $(X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ between objects is given by a triple $(f, g, \varphi )$ where $f : X' \to X$ and $g : S' \to S$ are morphisms of schemes which fit into a commutative diagram

\[ \xymatrix{ X' \ar[d] \ar[r]_ f & X \ar[d] \\ S' \ar[r]^ g & S } \]

inducing an isomorphism $X' \to S' \times _ S X$, in other words, the diagram is cartesian, and $\varphi : f^*\mathcal{L} \to \mathcal{L}'$ is an isomorphism. Composition is defined in the obvious manner (see Examples of Stacks, Sections 95.7 and 95.4). The forgetful functor

\[ p : \mathcal{P}\! \mathit{olarized}\longrightarrow \mathit{Sch}_{fppf},\quad (X \to S, \mathcal{L}) \longmapsto S \]

is how we view $\mathcal{P}\! \mathit{olarized}$ as a category over $\mathit{Sch}_{fppf}$ (see Section 99.2 for notation).

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