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The Stacks project

Lemma 71.5.6. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{F} be a finite type, quasi-coherent \mathcal{O}_ X-module. The closed subspaces

X = Z_{-1} \supset Z_0 \supset Z_1 \supset Z_2 \ldots

defined by the Fitting ideals of \mathcal{F} have the following properties

  1. The intersection \bigcap Z_ r is empty.

  2. The functor (\mathit{Sch}/X)^{opp} \to \textit{Sets} defined by the rule

    T \longmapsto \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ is locally generated by } \leq r\text{ sections} \\ \emptyset & \text{otherwise} \end{matrix} \right.

    is representable by the open subspace X \setminus Z_ r.

  3. The functor F_ r : (\mathit{Sch}/X)^{opp} \to \textit{Sets} defined by the rule

    T \longmapsto \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ locally free rank }r \\ \emptyset & \text{otherwise} \end{matrix} \right.

    is representable by the locally closed subspace Z_{r - 1} \setminus Z_ r of X.

If \mathcal{F} is of finite presentation, then Z_ r \to X, X \setminus Z_ r \to X, and Z_{r - 1} \setminus Z_ r \to X are of finite presentation.

Proof. Reduces to Divisors, Lemma 31.9.6 by étale localization. \square


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