The Stacks project

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

\[ K'_0(Z) = K_0(\textit{Coh}(Z)) \longrightarrow K_0(\textit{Coh}_ Z(X)) \]

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

\[ \ldots \subset \mathcal{F}^ p = \mathcal{I}^ p \mathcal{F} \subset \mathcal{F}^{p - 1} \subset \ldots \subset \mathcal{F}^0 = \mathcal{F} \]

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

\[ [\mathcal{F}] = [\bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}] \]

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

\[ \mathcal{F} \longmapsto c(\mathcal{F}) = [\bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}] \]

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

\[ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0 \]

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

\[ 0 \to (\mathcal{F} \cap \mathcal{I}^ q\mathcal{G})/ (\mathcal{F} \cap \mathcal{I}^{q + 1}\mathcal{G}) \to \mathcal{G}^ q/\mathcal{G}^{q + 1} \to \mathcal{H}^ q/\mathcal{H}^{q + 1} \to 0 \]

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

\[ \mathcal{F}^{p, q} = \mathcal{I}^ p\mathcal{F} \cap \mathcal{I}^ q\mathcal{G} \]

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

\[ [\bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p/\mathcal{F}^{p + 1}] = [\bigoplus \nolimits _{p, q \geq 0} \mathcal{F}^{p, q}/(\mathcal{F}^{p + 1, q} + \mathcal{F}^{p, q + 1})] = [\bigoplus \nolimits _{q \geq 0} (\mathcal{F} \cap \mathcal{I}^ q\mathcal{G})/ (\mathcal{F} \cap \mathcal{I}^{q + 1}\mathcal{G})] \]

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

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