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Support of a finite-type quasi-coherent sheaf is closed

Statement

Assume the Axiom of Choice, inherited from the construction of associated sheaves and from the stalk identification used below. Let $X$ be a scheme and let $F$ be a quasi-coherent $O_{X}$ -module of finite type (Finite type and finitely presented module sheaves), with support $Supp (F) = {x \in X : F_{x} \neq 0}$ (Support of a module sheaf).

Then $Supp (F)$ is a closed subset of $X$ . Moreover, if $U = Spec A$ is an affine open with $F ∣_{U} ≅ \tilde{M}$ for a finitely generated $A$ -module $M$ , then $Supp (F) \cap U = {p \in Spec A : {Ann}_{A} (M) \subseteq p} = V ({Ann}_{A} (M)),$ the zero set of the annihilator ideal of $M$ ; in particular $Supp (F) \cap U$ is closed in $U$ , and $F ∣_{U} = 0$ exactly when $M = 0$ .

Facts & Assumptions

Given: The Axiom of Choice; a scheme $X$ ; a finite-type quasi-coherent $O_{X}$ -module $F$ .

[F1]

The support is $Supp (F) = {x : F_{x} \neq 0}$ , and for an open $U \subseteq X$ one has $Supp (F ∣_{U}) = Supp (F) \cap U$ (Support of a module sheaf).

[F2]

For an affine scheme $Spec A$ with associated sheaf $\tilde{M}$ one has $(\tilde{M})_{p} ≅ M_{p}$ for every prime $p$ (The stalk of an associated sheaf is the localisation).

[F3]

If $M$ is a finitely generated $A$ -module, then ${Supp}_{A} (M) = {p : {Ann}_{A} (M) \subseteq p}$ (For a finite module, support is the set of primes containing the annihilator).

[F4]

$F$ is of finite type: every point of $X$ has an affine open neighbourhood $U = Spec A$ with $F ∣_{U} ≅ \tilde{M}$ for a finitely generated $A$ -module $M$ (Finite type and finitely presented module sheaves).

[F5]

A subset $Z \subseteq X$ is closed exactly when its complement is open, and openness of a subset can be checked on the members of any open cover: if every $W \cap U_{i}$ is open in $U_{i}$ for a cover $X = ⋃_{i} U_{i}$ by open sets, then $W$ is open in $X$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

[F6]

Axiom of Choice: every family of nonempty sets has a choice function (The Axiom of Choice).

Proof

Proof technique: direct; compute the support on affine charts and use that closedness can be checked on an open cover.

1.1F1F2F3

Let $U = Spec A$ be an affine open with $F ∣_{U} ≅ \tilde{M}$ for a finitely generated $A$ -module $M$ ; then $Supp (F) \cap U = Supp (F ∣_{U})$ by [F1] equals ${p : (\tilde{M})_{p} \neq 0}$ , which by [F2] is ${p : M_{p} \neq 0} = {Supp}_{A} (M)$ , and by [F3] this is ${p : {Ann}_{A} (M) \subseteq p} = V ({Ann}_{A} (M))$ , the zero set of an ideal and hence closed in $U = Spec A$ .

1.2F4given

Let ${U_{i}}_{i \in I}$ be the family of all affine open subsets $U_{i}$ of $X$ with $F ∣_{U_{i}} ≅ \tilde{M_{i}}$ for a finitely generated module $M_{i}$ ; by [F4] every point of $X$ lies in such a chart, so this family covers $X$ , and it is the family of all such charts, determined without selecting anything.

2.1F5step 1.1step 1.2

For every chart $U$ of the cover of step 1.2 the intersection $Supp (F) \cap U$ is closed in $U$ by step 1.1, so its complement $U ∖ Supp (F)$ is open in $U$ , and since the $U$ cover $X$ the complement $X ∖ Supp (F) = ⋃_{U} (U ∖ Supp (F))$ is open in $X$ by [F5]; hence $Supp (F)$ is closed in $X$ , as claimed.

3.1F2F3F6step 1.1step 2.1∎

The identification of the affine support with $V ({Ann}_{A} (M))$ is step 1.1, and the Axiom of Choice is used only through [F2], inherited from the associated-sheaf construction, and through the supplier [F3]; the cover of step 1.2 is the family of all admissible charts, so no selection occurs there.

Support of a finite-type quasi-coherent sheaf is closed

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources