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. $\square$

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