The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05JU. Beware of the difference between the letter 'O' and the digit '0'.