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

1. For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third.

2. 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 $(Z \to X)_*\mathcal{I}$.

3. There exists some coherent sheaf $\mathcal{G}$ on $X$ such that

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

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

3. $\dim _{\kappa (\xi )} \mathcal{G}_\xi = 1$, and

4. 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

Typos in the paragraph beginning "Next, suppose that $\mathcal{I}'$ is another quasi-coherent sheaf of ideals $Z_0$.":

1. Should say "of ideals on $Z_0$."

2. Replace the exact sequence with

3. In order for $\mathcal{Q}$ and $\mathcal{Q}'$ to have support strictly contained in $Z_0$, it looks like you should require $\mathcal{I}' \neq 0$ (if $\mathcal{I}' = 0$ the result is trivial anyway).

4. In "Hence $\mathcal{Q}$ and $\mathcal{Q}$ have property $\mathcal{P}$," the second $\mathcal{Q}$ should be $\mathcal{Q}'$.

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