30.12 Devissage of coherent sheaves
Let X be a Noetherian scheme. Consider an integral closed subscheme i : Z \to X. It is often convenient to consider coherent sheaves of the form i_*\mathcal{G} where \mathcal{G} is a coherent sheaf on Z. In particular we are interested in these sheaves when \mathcal{G} is a torsion free rank 1 sheaf. For example \mathcal{G} could be a nonzero sheaf of ideals on Z, or even more specifically \mathcal{G} = \mathcal{O}_ Z.
Throughout this section we will use that a coherent sheaf is the same thing as a finite type quasi-coherent sheaf and that a quasi-coherent subquotient of a coherent sheaf is coherent, see Section 30.9. The support of a coherent sheaf is closed, see Modules, Lemma 17.9.6.
Lemma 30.12.1. Let X be a Noetherian scheme. Let \mathcal{F} be a coherent sheaf on X. Suppose that \text{Supp}(\mathcal{F}) = Z \cup Z' with Z, Z' closed. Then there exists a short exact sequence of coherent sheaves
0 \to \mathcal{G}' \to \mathcal{F} \to \mathcal{G} \to 0
with \text{Supp}(\mathcal{G}') \subset Z' and \text{Supp}(\mathcal{G}) \subset Z.
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
0 \to \mathcal{G}'_ n \to \mathcal{F} \to \mathcal{G}_ n \to 0
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
Lemma 30.12.2. Let X be a Noetherian scheme. Let i : Z \to X be an integral closed subscheme. Let \xi \in Z be the generic point. Let \mathcal{F} be a coherent sheaf on X. Assume that \mathcal{F}_\xi is annihilated by \mathfrak m_\xi . Then there exist an integer r \geq 0 and a coherent sheaf of ideals \mathcal{I} \subset \mathcal{O}_ Z and an injective map of coherent sheaves
i_*\left(\mathcal{I}^{\oplus r}\right) \to \mathcal{F}
which is an isomorphism in a neighbourhood of \xi .
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 Y \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
Lemma 30.12.3. Let X be a Noetherian scheme. Let \mathcal{F} be a coherent sheaf on X. There exists a filtration
0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ m = \mathcal{F}
by coherent subsheaves such that for each j = 1, \ldots , m there exist an integral closed subscheme Z_ j \subset X and a nonzero coherent sheaf of ideals \mathcal{I}_ j \subset \mathcal{O}_{Z_ j} such that
\mathcal{F}_ j/\mathcal{F}_{j - 1} \cong (Z_ j \to X)_* \mathcal{I}_ j
Proof.
Consider the collection
\mathcal{T} = \left\{ \begin{matrix} Z \subset X \text{ closed such that there exists a coherent sheaf } \mathcal{F}
\\ \text{ with } \text{Supp}(\mathcal{F}) = Z \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 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
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 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
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. 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
0 \to i_*(\mathcal{I}^{\oplus r}) \to \mathcal{F} \to \mathcal{Q} \to 0
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
Lemma 30.12.4. Let X be a Noetherian scheme. Let \mathcal{P} be a property of coherent sheaves on X. Assume
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}.
For every integral closed subscheme Z \subset X and every quasi-coherent sheaf of ideals \mathcal{I} \subset \mathcal{O}_ Z we have \mathcal{P} for i_*\mathcal{I}.
Then property \mathcal{P} holds for every coherent sheaf on X.
Proof.
First 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 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
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
Lemma 30.12.6. Let X be a Noetherian scheme. Let \mathcal{P} be a property of coherent sheaves on X 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 X with generic point \xi there exists some coherent sheaf \mathcal{G} such that
\text{Supp}(\mathcal{G}) = Z,
\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 on X.
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
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
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}.
If \mathcal{P} holds for \mathcal{F}^{\oplus r} for some r \geq 1, then it holds for \mathcal{F}.
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} such that
\text{Supp}(\mathcal{G}) = Z_0,
\mathcal{G}_\xi is annihilated by \mathfrak m_\xi , and
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
Lemma 30.12.8. Let X be a Noetherian scheme. Let \mathcal{P} be a property of coherent sheaves on X such that
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}.
If \mathcal{P} holds for \mathcal{F}^{\oplus r} for some r \geq 1, then it holds for \mathcal{F}.
For every integral closed subscheme Z \subset X with generic point \xi there exists some coherent sheaf \mathcal{G} such that
\text{Supp}(\mathcal{G}) = Z,
\mathcal{G}_\xi is annihilated by \mathfrak m_\xi , and
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 on X.
Proof.
Follows from Lemma 30.12.7 in exactly the same way that Lemma 30.12.6 follows from Lemma 30.12.5.
\square
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