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.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05P8. Beware of the difference between the letter 'O' and the digit '0'.