The Stacks project

Remark 42.56.7. Let $X$ be a locally Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Consider the strictly full, saturated, triangulated subcategory

\[ D_{Z, perf}(\mathcal{O}_ X) \subset D(\mathcal{O}_ X) \]

consisting of perfect complexes of $\mathcal{O}_ X$-modules whose cohomology sheaves are settheoretically supported on $Z$. 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 $Z$. Observe that given $E \in D_{Z, perf}(\mathcal{O}_ X)$ Zariski locally on $X$ only a finite number of the cohomology sheaves $H^ i(E)$ are nonzero (and they are all settheoretically supported on $Z$). Hence we can define

\[ K_0(D_{Z, perf}(\mathcal{O}_ X)) \longrightarrow K_0(\textit{Coh}_ Z(X)) = K'_0(Z) \]

(equality by Lemma 42.23.6) by the rule

\[ E \longmapsto [\bigoplus \nolimits _{i \in \mathbf{Z}} H^{2i}(E)] - [\bigoplus \nolimits _{i \in \mathbf{Z}} H^{2i + 1}(E)] \]

This works because given a distinguished triangle in $D_{Z, perf}(\mathcal{O}_ X)$ we have a long exact sequence of cohomology sheaves.

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