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70.5 Fitting ideals

This section is the continuation of the discussion in Divisors, Section 31.9. 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. In this situation we can construct the Fitting ideals

\[ 0 = \text{Fit}_{-1}(\mathcal{F}) \subset \text{Fit}_0(\mathcal{F}) \subset \text{Fit}_1(\mathcal{F}) \subset \ldots \subset \mathcal{O}_ X \]

as the sequence of quasi-coherent sheaves ideals characterized by the following property: for every affine $U = \mathop{\mathrm{Spec}}(A)$ étale over $X$ if $\mathcal{F}|_ U$ corresponds to the $A$-module $M$, then $\text{Fit}_ i(\mathcal{F})|_ U$ corresponds to the ideal $\text{Fit}_ i(M) \subset A$. This is well defined and a quasi-coherent sheaf of ideals because if $A \to B$ is an étale ring map, then the $i$th Fitting ideal of $M \otimes _ A B$ over $B$ is equal to $\text{Fit}_ i(M) B$ by More on Algebra, Lemma 15.8.4 part (3). More precisely (perhaps), the existence of the quasi-coherent sheaves of ideals $\text{Fit}_0(\mathcal{O}_ X)$ follows (for example) from the description of quasi-coherent sheaves in Properties of Spaces, Lemma 65.29.3 and the pullback property given in Divisors, Lemma 31.9.1.

The advantage of constructing the Fitting ideals in this way is that we see immediately that formation of Fitting ideals commutes with étale localization hence many properties of the Fitting ideals immediately reduce to the corresponding properties in the case of schemes. Often we will use the discussion in Properties of Spaces, Section 65.30 to do the translation between properties of quasi-coherent sheaves on schemes and on algebraic spaces.

Lemma 70.5.1. 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 $f^{-1}\text{Fit}_ i(\mathcal{F}) \cdot \mathcal{O}_ X = \text{Fit}_ i(f^*\mathcal{F})$.

Proof. Reduces to Divisors, Lemma 31.9.1 by étale localization. $\square$

Lemma 70.5.2. 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.

Proof. Reduces to Divisors, Lemma 31.9.2 by étale localization. $\square$

Lemma 70.5.3. 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

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

Proof. Recall that formation of $Z$ commutes with étale localization, see Morphisms of Spaces, Definition 66.15.4 (which uses Morphisms of Spaces, Lemma 66.15.3 to define $Z$). Hence (1) and (2) follow from the case of schemes, see Divisors, Lemma 31.9.3. To get $\mathcal{G}_0$ as in part (3) we can use that we have $\mathcal{G}$ on $Z$ as in Morphisms of Spaces, Lemma 66.15.3 and set $\mathcal{G}_0 = (Z \to Z_0)_*\mathcal{G}$. $\square$

Lemma 70.5.4. 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 $\text{Fit}_ r(\mathcal{F})_{\overline{x}} = \mathcal{O}_{X, \overline{x}}$.

Proof. Reduces to Divisors, Lemma 31.9.4 by étale localization (as well as the description of the local ring in Properties of Spaces, Section 65.22 and the fact that the strict henselization of a local ring is faithfully flat to see that the equality over the strict henselization is equivalent to the equality over the local ring). $\square$

Lemma 70.5.5. 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 $k \geq r$.

Proof. Reduces to Divisors, Lemma 31.9.5 by étale localization. $\square$

Lemma 70.5.6. 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. $\square$

Lemma 70.5.7. 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 70.5.6. Set $X_ r = Z_{r - 1} \setminus Z_ r$. Then $X' = \coprod _{r \geq 0} X_ r$ represents the functor

\[ F_{flat} : \mathit{Sch}/X \longrightarrow \textit{Sets},\quad \quad T \longmapsto \left\{ \begin{matrix} \{ *\} & \text{if }\mathcal{F}_ T\text{ flat over }T \\ \emptyset & \text{otherwise} \end{matrix} \right. \]

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.

Proof. Reduces to Divisors, Lemma 31.9.7 by étale localization. $\square$


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