The Stacks project

Lemma 31.9.6. Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ S$-module. The closed subschemes

\[ S = Z_{-1} \supset Z_0 \supset Z_1 \supset Z_2 \ldots \]

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

  1. The intersection $\bigcap Z_ r$ is empty.

  2. The functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ 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 subscheme $S \setminus Z_ r$.

  3. The functor $F_ r : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$ 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 subscheme $Z_{r - 1} \setminus Z_ r$ of $S$.

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

Proof. Part (1) is true because over every affine open $U$ there is an integer $n$ such that $\text{Fit}_ n(\mathcal{F})|_ U = \mathcal{O}_ U$. Namely, we can take $n$ to be the number of generators of $\mathcal{F}$ over $U$, see More on Algebra, Section 15.8.

For any morphism $g : T \to S$ we see from Lemmas 31.9.1 and 31.9.4 that $\mathcal{F}_ T$ is locally generated by $\leq r$ sections if and only if $\text{Fit}_ r(\mathcal{F}) \cdot \mathcal{O}_ T = \mathcal{O}_ T$. This proves (2).

For any morphism $g : T \to S$ we see from Lemmas 31.9.1 and 31.9.5 that $\mathcal{F}_ T$ is free of rank $r$ if and only if $\text{Fit}_ r(\mathcal{F}) \cdot \mathcal{O}_ T = \mathcal{O}_ T$ and $\text{Fit}_{r - 1}(\mathcal{F}) \cdot \mathcal{O}_ T = 0$. This proves (3).

Assume $\mathcal{F}$ is of finite presentation. Then each of the morphisms $Z_ r \to S$ is of finite presentation as $\text{Fit}_ r(\mathcal{F})$ is of finite type (Lemma 31.9.2 and Morphisms, Lemma 29.21.7). This implies that $Z_{r - 1} \setminus Z_ r$ is a retrocompact open in $Z_ r$ (Properties, Lemma 28.24.1) and hence the morphism $Z_{r - 1} \setminus Z_ r \to Z_ r$ is of finite presentation as well. $\square$

Comments (2)

Comment #7940 by on

Seemingly the last part should be put into the enumerate-environment since you refer to it as (4) in the proof.

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