The Stacks project

Remark 36.38.9. Let $X$ be a 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 $Z$. The zeroth $K$-group $K_0(D_{Z, perf}(\mathcal{O}_ X))$ of this triangulated category is sometimes denoted $K_ Z(X)$ or $K_{0, Z}(X)$. Using derived tensor product exactly as in Remark 36.38.6 we see that $K_0(D_{Z, perf}(\mathcal{O}_ X))$ has a multiplication which is associative and commutative, but in general $K_0(D_{Z, perf}(\mathcal{O}_ X))$ doesn't have a unit.

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