The Stacks project

Lemma 69.14.5. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$. Assume

  1. For any short exact sequence of coherent sheaves

    \[ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 \]

    if $\mathcal{F}_ i$, $i = 1, 2$ have property $\mathcal{P}$ then so does $\mathcal{F}$.

  2. If $\mathcal{P}$ holds for $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$.

  3. For every reduced closed subspace $i : Z \to X$ with $|Z|$ irreducible there exists a coherent sheaf $\mathcal{G}$ on $Z$ such that

    1. $\text{Supp}(\mathcal{G}) = Z$,

    2. for every nonzero quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Z$ there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{I}\mathcal{G}$ such that $\text{Supp}(\mathcal{G}/\mathcal{G}')$ is proper closed in $|Z|$ and such that $\mathcal{P}$ holds for $i_*\mathcal{G}'$.

Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.

Proof. Consider the collection

\[ \mathcal{T} = \left\{ \begin{matrix} T \subset |X| \text{ nonempty closed such that there exists a coherent sheaf } \\ \mathcal{F} \text{ with } \text{Supp}(\mathcal{F}) = T \text{ for which the lemma is wrong} \end{matrix} \right\} \]

We are trying to show that $\mathcal{T}$ is empty. If not, then because $|X|$ is Noetherian (Properties of Spaces, Lemma 66.24.2) we can choose a minimal element $T \in \mathcal{T}$. This means that there exists a coherent sheaf $\mathcal{F}$ on $X$ whose support is $T$ and for which the lemma does not hold.

If $T$ is not irreducible, then we can write $T = Z_1 \cup Z_2$ with $Z_1, Z_2$ closed and strictly smaller than $T$. Then we can apply Lemma 69.14.1 to get a short exact sequence of coherent sheaves

\[ 0 \to \mathcal{G}_1 \to \mathcal{F} \to \mathcal{G}_2 \to 0 \]

with $\text{Supp}(\mathcal{G}_ i) \subset Z_ i$. By minimality of $T$ each of $\mathcal{G}_ i$ has $\mathcal{P}$. Hence $\mathcal{F}$ has property $\mathcal{P}$ by (1), a contradiction.

Suppose $T$ is irreducible. Let $\mathcal{J}$ be the sheaf of ideals defining the reduced induced closed subspace structure on $T$, see Properties of Spaces, Lemma 66.12.3. By Lemma 69.13.2 we see there exists an $n \geq 0$ such that $\mathcal{J}^ n\mathcal{F} = 0$. Hence we obtain a filtration

\[ 0 = \mathcal{J}^ n\mathcal{F} \subset \mathcal{J}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{J}\mathcal{F} \subset \mathcal{F} \]

each of whose successive subquotients is annihilated by $\mathcal{J}$. Hence if each of these subquotients has a filtration as in the statement of the lemma then also $\mathcal{F}$ does by (1). In other words we may assume that $\mathcal{J}$ does annihilate $\mathcal{F}$.

Assume $T$ is irreducible and $\mathcal{J}\mathcal{F} = 0$ where $\mathcal{J}$ is as above. Denote $i : Z \to X$ the closed subspace corresponding to $\mathcal{J}$. Then $\mathcal{F} = i_*\mathcal{H}$ for some coherent $\mathcal{O}_ Z$-module $\mathcal{H}$, see Morphisms of Spaces, Lemma 67.14.1 and Lemma 69.12.7. Let $\mathcal{G}$ be the coherent sheaf on $Z$ satisfying (3)(a) and (3)(b). We apply Lemma 69.14.2 to get injective maps

\[ \mathcal{I}_1^{\oplus r_1} \to \mathcal{H} \quad \text{and}\quad \mathcal{I}_2^{\oplus r_2} \to \mathcal{G} \]

where the support of the cokernels are proper closed in $Z$. Hence we find an nonempty open $V \subset Z$ such that

\[ \mathcal{H}^{\oplus r_2}_ V \cong \mathcal{G}^{\oplus r_1}_ V \]

Let $\mathcal{I} \subset \mathcal{O}_ Z$ be a quasi-coherent ideal sheaf cutting out $Z \setminus V$ we obtain (Lemma 69.13.4) a map

\[ \mathcal{I}^ n\mathcal{G}^{\oplus r_1} \longrightarrow \mathcal{H}^{\oplus r_2} \]

which is an isomorphism over $V$. The kernel is supported on $Z \setminus V$ hence annihilated by some power of $\mathcal{I}$, see Lemma 69.13.2. Thus after increasing $n$ we may assume the displayed map is injective, see Lemma 69.13.3. Applying (3)(b) we find $\mathcal{G}' \subset \mathcal{I}^ n\mathcal{G}$ such that

\[ (i_*\mathcal{G}')^{\oplus r_1} \longrightarrow i_*\mathcal{H}^{\oplus r_2} = \mathcal{F}^{\oplus r_2} \]

is injective with cokernel supported in a proper closed subset of $Z$ and such that property $\mathcal{P}$ holds for $i_*\mathcal{G}'$. By (1) property $\mathcal{P}$ holds for $(i_*\mathcal{G}')^{\oplus r_1}$. By (1) and minimality of $T = |Z|$ property $\mathcal{P}$ holds for $\mathcal{F}^{\oplus r_2}$. And finally by (2) property $\mathcal{P}$ holds for $\mathcal{F}$ which is the desired contradiction. $\square$

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