The Stacks project

Lemma 31.10.1. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $X = Z_{-1} \supset Z_0 \supset Z_1 \supset \ldots $ be the closed subschemes defined by the fitting ideals of $\Omega _{X/S}$. Then the formation of $Z_ i$ commutes with arbitrary base change.

Proof. Observe that $\Omega _{X/S}$ is a finite type quasi-coherent $\mathcal{O}_ X$-module (Morphisms, Lemma 29.32.12) hence the fitting ideals are defined. If $f' : X' \to S'$ is the base change of $f$ by $g : S' \to S$, then $\Omega _{X'/S'} = (g')^*\Omega _{X/S}$ where $g' : X' \to X$ is the projection (Morphisms, Lemma 29.32.10). Hence $(g')^{-1}\text{Fit}_ i(\Omega _{X/S}) \cdot \mathcal{O}_{X'} = \text{Fit}_ i(\Omega _{X'/S'})$. This means that

\[ Z'_ i = (g')^{-1}(Z_ i) = Z_ i \times _ X X' \]

scheme theoretically and this is the meaning of the statement of the lemma. $\square$

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