The Stacks project

Proof. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the sheaf of ideals defining the reduced induced closed subscheme structure on $Z$, see Schemes, Lemma 26.12.4. Consider the subsheaves $\mathcal{G}'_ n = \mathcal{I}^ n\mathcal{F}$ and the quotients $\mathcal{G}_ n = \mathcal{F}/\mathcal{I}^ n\mathcal{F}$. For each $n$ we have a short exact sequence

For every point $x$ of $Z' \setminus Z$ we have $\mathcal{I}_ x = \mathcal{O}_{X, x}$ and hence $\mathcal{G}_{n, x} = 0$. Thus we see that $\text{Supp}(\mathcal{G}_ n) \subset Z$. Note that $X \setminus Z'$ is a Noetherian scheme. Hence by Lemma 30.10.2 there exists an $n$ such that $\mathcal{G}'_ n|_{X \setminus Z'} = \mathcal{I}^ n\mathcal{F}|_{X \setminus Z'} = 0$. For such an $n$ we see that $\text{Supp}(\mathcal{G}'_ n) \subset Z'$. Thus setting $\mathcal{G}' = \mathcal{G}'_ n$ and $\mathcal{G} = \mathcal{G}_ n$ works. $\square$

Proof. Let $\mathcal{J} \subset \mathcal{O}_ X$ be the ideal sheaf of $Z$. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of local sections of $\mathcal{F}$ which are annihilated by $\mathcal{J}$. It is a quasi-coherent sheaf by Properties, Lemma 28.24.2. Moreover, $\mathcal{F}'_\xi = \mathcal{F}_\xi $ because $\mathcal{J}_\xi = \mathfrak m_\xi $ and part (3) of Properties, Lemma 28.24.2. By Lemma 30.9.5 we see that $\mathcal{F}' \to \mathcal{F}$ induces an isomorphism in a neighbourhood of $\xi $. Hence we may replace $\mathcal{F}$ by $\mathcal{F}'$ and assume that $\mathcal{F}$ is annihilated by $\mathcal{J}$.

Assume $\mathcal{J}\mathcal{F} = 0$. By Lemma 30.9.8 we can write $\mathcal{F} = i_*\mathcal{G}$ for some coherent sheaf $\mathcal{G}$ on $Z$. Suppose we can find a morphism $\mathcal{I}^{\oplus r} \to \mathcal{G}$ which is an isomorphism in a neighbourhood of the generic point $\xi $ of $Z$. Then applying $i_*$ (which is left exact) we get the result of the lemma. Hence we have reduced to the case $X = Z$.

Suppose $Z = X$ is an integral Noetherian scheme with generic point $\xi $. Note that $\mathcal{O}_{X, \xi } = \kappa (\xi )$ is the function field of $X$ in this case. Since $\mathcal{F}_\xi $ is a finite $\mathcal{O}_\xi $-module we see that $r = \dim _{\kappa (\xi )} \mathcal{F}_\xi $ is finite. Hence the sheaves $\mathcal{O}_ X^{\oplus r}$ and $\mathcal{F}$ have isomorphic stalks at $\xi $. By Lemma 30.9.6 there exists a nonempty open $U \subset X$ and a morphism $\psi : \mathcal{O}_ X^{\oplus r}|_ U \to \mathcal{F}|_ U$ which is an isomorphism at $\xi $, and hence an isomorphism in a neighbourhood of $\xi $ by Lemma 30.9.5. By Schemes, Lemma 26.12.4 there exists a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ whose associated closed subscheme $Z \subset X$ is the complement of $U$. By Lemma 30.10.5 there exists an $n \geq 0$ and a morphism $\mathcal{I}^ n(\mathcal{O}_ X^{\oplus r}) \to \mathcal{F}$ which recovers our $\psi $ over $U$. Since $\mathcal{I}^ n(\mathcal{O}_ X^{\oplus r}) = (\mathcal{I}^ n)^{\oplus r}$ we get a map as in the lemma. It is injective because $X$ is integral and it is injective at the generic point of $X$ (easy proof omitted). $\square$

Proof. Consider the collection

We are trying to show that $\mathcal{T}$ is empty. If not, then because $X$ is Noetherian we can choose a minimal element $Z \in \mathcal{T}$. This means that there exists a coherent sheaf $\mathcal{F}$ on $X$ whose support is $Z$ and for which the lemma does not hold. Clearly $Z \not= \emptyset $ since the only sheaf whose support is empty is the zero sheaf for which the lemma does hold (with $m = 0$).

If $Z$ is not irreducible, then we can write $Z = Z_1 \cup Z_2$ with $Z_1, Z_2$ closed and strictly smaller than $Z$. Then we can apply Lemma 30.12.1 to get a short exact sequence of coherent sheaves

with $\text{Supp}(\mathcal{G}_ i) \subset Z_ i$. By minimality of $Z$ each of $\mathcal{G}_ i$ has a filtration as in the statement of the lemma. By considering the induced filtration on $\mathcal{F}$ we arrive at a contradiction. Hence we conclude that $Z$ is irreducible.

Suppose $Z$ is irreducible. Let $\mathcal{J}$ be the sheaf of ideals cutting out the reduced induced closed subscheme structure of $Z$, see Schemes, Lemma 26.12.4. By Lemma 30.10.2 we see there exists an $n \geq 0$ such that $\mathcal{J}^ n\mathcal{F} = 0$. Hence we obtain a filtration

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. In other words we may assume that $\mathcal{J}$ does annihilate $\mathcal{F}$.

In the case where $Z$ is irreducible and $\mathcal{J}\mathcal{F} = 0$ we can apply Lemma 30.12.2. This gives a short exact sequence

where $\mathcal{Q}$ is defined as the quotient. Since $\mathcal{Q}$ is zero in a neighbourhood of $\xi $ by the lemma just cited we see that the support of $\mathcal{Q}$ is strictly smaller than $Z$. Hence we see that $\mathcal{Q}$ has a filtration of the desired type by minimality of $Z$. But then clearly $\mathcal{F}$ does too, which is our final contradiction. $\square$

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

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 the property (1) for $\mathcal{P}$. On the other hand, by Lemma 30.12.3 we can filter any $\mathcal{F}$ with successive subquotients as in (2). Hence the lemma follows. $\square$

Proof. First note that if $\mathcal{F}$ is a coherent sheaf with support contained in $Z_0$ with a filtration

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

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

and

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. $\square$

Proof. According to Lemma 30.12.4 it suffices to show that for all integral closed subschemes $Z \subset X$ and all quasi-coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}_ Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. If this fails, then since $X$ is Noetherian there is a minimal integral closed subscheme $Z_0 \subset X$ such that $\mathcal{P}$ fails for $(Z_0 \to X)_*\mathcal{I}_0$ for some quasi-coherent sheaf of ideals $\mathcal{I}_0 \subset \mathcal{O}_{Z_0}$, but $\mathcal{P}$ does hold for $(Z \to X)_*\mathcal{I}$ for all integral closed subschemes $Z \subset Z_0$, $Z \not= Z_0$ and quasi-coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}_ Z$. Since we have the existence of $\mathcal{G}$ for $Z_0$ by part (2), according to Lemma 30.12.5 this cannot happen. $\square$

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

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

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

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

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

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. $\square$

Proof. Follows from Lemma 30.12.7 in exactly the same way that Lemma 30.12.6 follows from Lemma 30.12.5. $\square$