The Stacks project

Definition 87.9.1. Let $S$ be a scheme. We say a sheaf $X$ on $(\mathit{Sch}/S)_{fppf}$ is an affine formal algebraic space if there exist

  1. a directed set $\Lambda $,

  2. a system $(X_\lambda , f_{\lambda \mu })$ over $\Lambda $ in $(\mathit{Sch}/S)_{fppf}$ where

    1. each $X_\lambda $ is affine,

    2. each $f_{\lambda \mu } : X_\lambda \to X_\mu $ is a thickening,

such that

\[ X \cong \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } X_\lambda \]

as fppf sheaves and $X$ satisfies a set theoretic condition (see Remark 87.11.5). A morphism of affine formal algebraic spaces over $S$ is a map of sheaves.

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View Definition 87.9.1 as pdf