Definition 75.9.1. Thickenings. Let $S$ be a scheme.

1. We say an algebraic space $X'$ is a thickening of an algebraic space $X$ if $X$ is a closed subspace of $X'$ and the associated topological spaces are equal.

2. We say $X'$ is a first order thickening of $X$ if $X$ is a closed subspace of $X'$ and the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_{X'}$ defining $X$ has square zero.

3. Given two thickenings $X \subset X'$ and $Y \subset Y'$ a morphism of thickenings is a morphism $f' : X' \to Y'$ such that $f(X) \subset Y$, i.e., such that $f'|_ X$ factors through the closed subspace $Y$. In this situation we set $f = f'|_ X : X \to Y$ and we say that $(f, f') : (X \subset X') \to (Y \subset Y')$ is a morphism of thickenings.

4. Let $B$ be an algebraic space. We similarly define thickenings over $B$, and morphisms of thickenings over $B$. This means that the spaces $X, X', Y, Y'$ above are algebraic spaces endowed with a structure morphism to $B$, and that the morphisms $X \to X'$, $Y \to Y'$ and $f' : X' \to Y'$ are morphisms over $B$.

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