The Stacks project

Definition 97.5.1. Let $S$ be a locally Noetherian scheme. Let $\mathcal{Z}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $\mathcal{Z}$ satisfies condition (RS) if for every pushout

\[ \xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' = Y \amalg _ X X' } \]

in the category of schemes over $S$ where

  1. $X$, $X'$, $Y$, $Y'$ are spectra of local Artinian rings,

  2. $X$, $X'$, $Y$, $Y'$ are of finite type over $S$, and

  3. $X \to X'$ (and hence $Y \to Y'$) is a closed immersion

the functor of fibre categories

\[ \mathcal{Z}_{Y'} \longrightarrow \mathcal{Z}_ Y \times _{\mathcal{Z}_ X} \mathcal{Z}_{X'} \]

is an equivalence of categories.

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