The Stacks project

Definition 36.38.2. Let $X$ be a scheme.

  1. We denote $K_0(X)$ the Grothendieck group of $X$. It is the zeroth K-group of the strictly full, saturated, triangulated subcategory $D_{perf}(\mathcal{O}_ X)$ of $D(\mathcal{O}_ X)$ consisting of perfect objects. In a formula

    \[ K_0(X) = K_0(D_{perf}(\mathcal{O}_ X)) \]

  2. If $X$ is locally Noetherian, then we denote $K'_0(X)$ the Grothendieck group of coherent sheaves on $X$. It is the is the zeroth $K$-group of the abelian category of coherent $\mathcal{O}_ X$-modules. In a formula

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

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