The Stacks project

Lemma 71.5.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ Y$-module. Then $f^{-1}\text{Fit}_ i(\mathcal{F}) \cdot \mathcal{O}_ X = \text{Fit}_ i(f^*\mathcal{F})$.

Lemma 71.5.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module. Then $\text{Fit}_ r(\mathcal{F})$ is a quasi-coherent ideal of finite type.

Lemma 71.5.3. 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. Let $Z_0 \subset X$ be the closed subspace cut out by $\text{Fit}_0(\mathcal{F})$. Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Then

1. $Z \subset Z_0 \subset X$ as closed subspaces,

2. $|Z| = |Z_0| = \text{Supp}(\mathcal{F})$ as closed subsets of $|X|$,

3. there exists a finite type, quasi-coherent $\mathcal{O}_{Z_0}$-module $\mathcal{G}_0$ with

   $$(Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}.$$

Lemma 71.5.4. 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. Let $x \in |X|$. Then $\mathcal{F}$ can be generated by $r$ elements in an étale neighbourhood of $x$ if and only if $\text{Fit}_ r(\mathcal{F})_{\overline{x}} = \mathcal{O}_{X, \overline{x}}$.

Lemma 71.5.5. 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. Let $r \geq 0$. The following are equivalent

1. $\mathcal{F}$ is finite locally free of rank $r$

2. $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and $\text{Fit}_ r(\mathcal{F}) = \mathcal{O}_ X$, and

3. $\text{Fit}_ k(\mathcal{F}) = 0$ for $k < r$ and $\text{Fit}_ k(\mathcal{F}) = \mathcal{O}_ X$ for $k \geq r$.

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

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.

Lemma 71.5.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation. Let $X = Z_{-1} \subset Z_0 \subset Z_1 \subset \ldots$ be as in Lemma 71.5.6. Set $X_ r = Z_{r - 1} \setminus Z_ r$. Then $X' = \coprod _{r \geq 0} X_ r$ represents the functor

Moreover, $\mathcal{F}|_{X_ r}$ is locally free of rank $r$ and the morphisms $X_ r \to X$ and $X' \to X$ are of finite presentation.