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 ├ętale localization. $\square$


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