Lemma 30.12.7. Let $X$ be a Noetherian scheme. Let $Z_0 \subset X$ be an irreducible closed subset with generic point $\xi$. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that

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 integral closed subscheme $Z \subset Z_0 \subset X$, $Z \not= Z_0$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$.

4. There exists some coherent sheaf $\mathcal{G}$ such that

1. $\text{Supp}(\mathcal{G}) = Z_0$,

2. $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and

3. for every quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ X$ such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi }$ there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi$ and such that $\mathcal{P}$ holds for $\mathcal{G}'$.

Then property $\mathcal{P}$ holds for every coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$.

Proof. Note that if $\mathcal{F}$ is a coherent sheaf with a filtration

$0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ m = \mathcal{F}$

by coherent subsheaves such that each of $\mathcal{F}_ i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$, then so does $\mathcal{F}$. This follows from assumption (1).

As a first application we conclude that any coherent sheaf whose support is strictly contained in $Z_0$ has property $\mathcal{P}$. Namely, such a sheaf has a filtration (see Lemma 30.12.3) whose subquotients have property $\mathcal{P}$ according to (3).

Let us denote $i : Z_0 \to X$ the closed immersion. Consider a coherent sheaf $\mathcal{G}$ as in (4). By Lemma 30.12.2 there exists a sheaf of ideals $\mathcal{I}$ on $Z_0$ and a short exact sequence

$0 \to i_*\mathcal{I}^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0$

where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. In particular $r > 0$ and $\mathcal{I}$ is nonzero because the support of $\mathcal{G}$ is equal to $Z_0$. Let $\mathcal{I}' \subset \mathcal{I}$ be any nonzero quasi-coherent sheaf of ideals on $Z_0$ contained in $\mathcal{I}$. Then we also get a short exact sequence

$0 \to i_*(\mathcal{I}')^{\oplus r} \to \mathcal{G} \to \mathcal{Q}' \to 0$

where $\mathcal{Q}'$ has support properly contained in $Z_0$. Let $\mathcal{J} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals cutting out the support of $\mathcal{Q}'$ (for example the ideal corresponding to the reduced induced closed subscheme structure on the support of $\mathcal{Q}'$). Then $\mathcal{J}_\xi = \mathcal{O}_{X, \xi }$. By Lemma 30.10.2 we see that $\mathcal{J}^ n\mathcal{Q}' = 0$ for some $n$. Hence $\mathcal{J}^ n\mathcal{G} \subset i_*(\mathcal{I}')^{\oplus r}$. By assumption (4)(c) of the lemma we see there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}^ n\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi$ for which property $\mathcal{P}$ holds. Hence we get a short exact sequence

$0 \to \mathcal{G}' \to i_*(\mathcal{I}')^{\oplus r} \to \mathcal{Q}'' \to 0$

where $\mathcal{Q}''$ has support properly contained in $Z_0$. Thus by our initial remarks and property (1) of the lemma we conclude that $i_*(\mathcal{I}')^{\oplus r}$ satisfies $\mathcal{P}$. Hence we see that $i_*\mathcal{I}'$ satisfies $\mathcal{P}$ by (2). Finally, for an arbitrary quasi-coherent sheaf of ideals $\mathcal{I}'' \subset \mathcal{O}_{Z_0}$ we can set $\mathcal{I}' = \mathcal{I}'' \cap \mathcal{I}$ and we get a short exact sequence

$0 \to i_*(\mathcal{I}') \to i_*(\mathcal{I}'') \to \mathcal{Q}''' \to 0$

where $\mathcal{Q}'''$ has support properly contained in $Z_0$. Hence we conclude that property $\mathcal{P}$ holds for $i_*\mathcal{I}''$ as well.

The final step of the proof is to note that any coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$ has a filtration (see Lemma 30.12.3 again) whose subquotients all have property $\mathcal{P}$ by what we just said. $\square$

Comment #952 by correction_bot on

In "Let $\mathcal{I}' \subset \mathcal{I}$ be any nonzero quasi-coherent sheaf of ideals on $Z$ contained in $\mathcal{I}$", $Z$ should be $Z_0$.

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