Lemma 42.23.6. Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Denote $\textit{Coh}_ Z(X) \subset \textit{Coh}(X)$ the Serre subcategory of coherent $\mathcal{O}_ X$-modules whose set theoretic support is contained in $Z$. Then the exact inclusion functor $\textit{Coh}(Z) \to \textit{Coh}_ Z(X)$ induces an isomorphism

**Proof.**
Let $\mathcal{F}$ be an object of $\textit{Coh}_ Z(X)$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent ideal sheaf of $Z$. Consider the descending filtration

Exactly as in the proof of Lemma 42.23.4 this filtration is locally finite and hence $\bigoplus _{p \geq 0} \mathcal{F}^ p$, $\bigoplus _{p \geq 1} \mathcal{F}^ p$, and $\bigoplus _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}$ are coherent $\mathcal{O}_ X$-modules supported on $Z$. Hence we get

in $K_0(\textit{Coh}_ Z(X))$ exactly as in the proof of Lemma 42.23.4. Since the coherent module $\bigoplus _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}$ is annihilated by $\mathcal{I}$ we conclude that $[\mathcal{F}]$ is in the image. Actually, we claim that the map

factors through $K_0(\textit{Coh}_ Z(X))$ and is an inverse to the map in the statement of the lemma. To see this all we have to show is that if

is a short exact sequence in $\textit{Coh}_ Z(X)$, then we get $c(\mathcal{G}) = c(\mathcal{F}) + c(\mathcal{H})$. Observe that for all $q \geq 0$ we have a short exact sequence

For $p, q \geq 0$ consider the coherent submodule

Arguing exactly as above and using that the filtrations $\mathcal{F}^ p = \mathcal{I}^ p\mathcal{F}$ and $\mathcal{F} \cap \mathcal{I}^ q\mathcal{G}$ are locally finite, we find that

in $K_0(\textit{Coh}(Z))$. Combined with the exact sequences above we obtain the desired result. Some details omitted. $\square$

## 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Your email address will not be published.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.

## Comments (0)