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

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

(equality by Lemma 42.23.6) by the rule

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