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:

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 $I = \text{Ann}_ A(M)$.

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

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

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

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