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29.5 Supports of modules

In this section we collect some elementary results on supports of quasi-coherent modules on schemes. Recall that the support of a sheaf of modules has been defined in Modules, Section 17.5. On the other hand, the support of a module was defined in Algebra, Section 10.62. These match.

Lemma 29.5.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathop{\mathrm{Spec}}(A) = U \subset X$ be an affine open, and set $M = \Gamma (U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. The following are equivalent

  1. $\mathfrak p$ is in the support of $M$, and

  2. $x$ is in the support of $\mathcal{F}$.

Proof. This follows from the equality $\mathcal{F}_ x = M_{\mathfrak p}$, see Schemes, Lemma 26.5.4 and the definitions. $\square$

Lemma 29.5.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. The support of $\mathcal{F}$ is closed under specialization.

Proof. If $x' \leadsto x$ is a specialization and $\mathcal{F}_ x = 0$ then $\mathcal{F}_{x'}$ is zero, as $\mathcal{F}_{x'}$ is a localization of the module $\mathcal{F}_ x$. Hence the complement of $\text{Supp}(\mathcal{F})$ is closed under generalization. $\square$

For finite type quasi-coherent modules the support is closed, can be checked on fibres, and commutes with base change.

Lemma 29.5.3. Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. Then

  1. The support of $\mathcal{F}$ is closed.

  2. For $x \in X$ we have

    \[ x \in \text{Supp}(\mathcal{F}) \Leftrightarrow \mathcal{F}_ x \not= 0 \Leftrightarrow \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \not= 0. \]
  3. For any morphism of schemes $f : Y \to X$ the pullback $f^*\mathcal{F}$ is of finite type as well and we have $\text{Supp}(f^*\mathcal{F}) = f^{-1}(\text{Supp}(\mathcal{F}))$.

Proof. Part (1) is a reformulation of Modules, Lemma 17.9.6. You can also combine Lemma 29.5.1, Properties, Lemma 28.16.1, and Algebra, Lemma 10.40.5 to see this. The first equivalence in (2) is the definition of support, and the second equivalence follows from Nakayama's lemma, see Algebra, Lemma 10.20.1. Let $f : Y \to X$ be a morphism of schemes. Note that $f^*\mathcal{F}$ is of finite type by Modules, Lemma 17.9.2. For the final assertion, let $y \in Y$ with image $x \in X$. Recall that

\[ (f^*\mathcal{F})_ y = \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \mathcal{O}_{Y, y}, \]

see Sheaves, Lemma 6.26.4. Hence $(f^*\mathcal{F})_ y \otimes \kappa (y)$ is nonzero if and only if $\mathcal{F}_ x \otimes \kappa (x)$ is nonzero. By (2) this implies $x \in \text{Supp}(\mathcal{F})$ if and only if $y \in \text{Supp}(f^*\mathcal{F})$, which is the content of assertion (3). $\square$

Lemma 29.5.4. Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. There exists a smallest closed subscheme $i : Z \to X$ such that there exists a quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ with $i_*\mathcal{G} \cong \mathcal{F}$. Moreover:

  1. If $\mathop{\mathrm{Spec}}(A) \subset X$ is any affine open, and $\mathcal{F}|_{\mathop{\mathrm{Spec}}(A)} = \widetilde{M}$ then $Z \cap \mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{Spec}}(A/I)$ where $I = \text{Ann}_ A(M)$.

  2. The quasi-coherent sheaf $\mathcal{G}$ is unique up to unique isomorphism.

  3. The quasi-coherent sheaf $\mathcal{G}$ is of finite type.

  4. The support of $\mathcal{G}$ and of $\mathcal{F}$ is $Z$.

Proof. Suppose that $i' : Z' \to X$ is a closed subscheme which satisfies the description on open affines from the lemma. Then by Lemma 29.4.1 we see that $\mathcal{F} \cong i'_*\mathcal{G}'$ for some unique quasi-coherent sheaf $\mathcal{G}'$ on $Z'$. Furthermore, it is clear that $Z'$ is the smallest closed subscheme with this property (by the same lemma). Finally, using Properties, Lemma 28.16.1 and Algebra, Lemma 10.5.5 it follows that $\mathcal{G}'$ is of finite type. We have $\text{Supp}(\mathcal{G}') = Z$ by Algebra, Lemma 10.40.5. Hence, in order to prove the lemma it suffices to show that the characterization in (1) actually does define a closed subscheme. And, in order to do this it suffices to prove that the given rule produces a quasi-coherent sheaf of ideals, see Lemma 29.2.3. This comes down to the following algebra fact: If $A$ is a ring, $f \in A$, and $M$ is a finite $A$-module, then $\text{Ann}_ A(M)_ f = \text{Ann}_{A_ f}(M_ f)$. We omit the proof. $\square$

Definition 29.5.5. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. The scheme theoretic support of $\mathcal{F}$ is the closed subscheme $Z \subset X$ constructed in Lemma 29.5.4.

In this situation we often think of $\mathcal{F}$ as a quasi-coherent sheaf of finite type on $Z$ (via the equivalence of categories of Lemma 29.4.1).

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