Lemma 30.12.5. 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 $X$ with support contained in $Z_0$ such that
For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third.
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 $(Z \to X)_*\mathcal{I}$.
There exists some coherent sheaf $\mathcal{G}$ on $X$ such that
$\text{Supp}(\mathcal{G}) = Z_0$,
$\mathcal{G}_\xi $ is annihilated by $\mathfrak m_\xi $,
$\dim _{\kappa (\xi )} \mathcal{G}_\xi = 1$, and
property $\mathcal{P}$ 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.
First note that if $\mathcal{F}$ is a coherent sheaf with support contained in $Z_0$ 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 $\mathcal{F}$. Or, if $\mathcal{F}$ has property $\mathcal{P}$ and all but one of the $\mathcal{F}_ i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$ then so does the last one. 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 (2).
Let $\mathcal{G}$ be as in (3). By Lemma 30.12.2 there exist a sheaf of ideals $\mathcal{I}$ on $Z_0$, an integer $r \geq 1$, and a short exact sequence
\[ 0 \to \left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0 \]
where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. By (3)(c) we see that $r = 1$. Since $\mathcal{Q}$ has property $\mathcal{P}$ too we conclude that $(Z_0 \to X)_*\mathcal{I}$ has property $\mathcal{P}$.
Next, suppose that $\mathcal{I}' \not= 0$ is another quasi-coherent sheaf of ideals on $Z_0$. Then we can consider the intersection $\mathcal{I}'' = \mathcal{I}' \cap \mathcal{I}$ and we get two short exact sequences
\[ 0 \to (Z_0 \to X)_*\mathcal{I}'' \to (Z_0 \to X)_*\mathcal{I} \to \mathcal{Q} \to 0 \]
and
\[ 0 \to (Z_0 \to X)_*\mathcal{I}'' \to (Z_0 \to X)_*\mathcal{I}' \to \mathcal{Q}' \to 0. \]
Note that the support of the coherent sheaves $\mathcal{Q}$ and $\mathcal{Q}'$ are strictly contained in $Z_0$. Hence $\mathcal{Q}$ and $\mathcal{Q}'$ have property $\mathcal{P}$ (see above). Hence we conclude using (1) that $(Z_0 \to X)_*\mathcal{I}''$ and $(Z_0 \to X)_*\mathcal{I}'$ both have $\mathcal{P}$ 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.
$\square$
Comments (1)
Comment #951 by correction_bot on
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