Definition 37.2.1. Thickenings.

1. We say a scheme $X'$ is a thickening of a scheme $X$ if $X$ is a closed subscheme of $X'$ and the underlying topological spaces are equal.

2. We say a scheme $X'$ is a first order thickening of a scheme $X$ if $X$ is a closed subscheme 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 subscheme $Y$. 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 $S$ be a scheme. We similarly define thickenings over $S$, and morphisms of thickenings over $S$. This means that the schemes $X, X', Y, Y'$ above are schemes over $S$, and that the morphisms $X \to X'$, $Y \to Y'$ and $f' : X' \to Y'$ are morphisms over $S$.

Comment #620 by Anfang on

There is a typo in definition 3): it should be $f'(X) \subset Y$, not $f(X) \subset Y$.

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