The Stacks project

75.2 Conventions

If $\mathcal{A}$ is an abelian category and $M$ is an object of $\mathcal{A}$ then we also denote $M$ the object of $K(\mathcal{A})$ and/or $D(\mathcal{A})$ corresponding to the complex which has $M$ in degree $0$ and is zero in all other degrees.

If we have a ring $A$, then $K(A)$ denotes the homotopy category of complexes of $A$-modules and $D(A)$ the associated derived category. Similarly, if we have a ringed space $(X, \mathcal{O}_ X)$ the symbol $K(\mathcal{O}_ X)$ denotes the homotopy category of complexes of $\mathcal{O}_ X$-modules and $D(\mathcal{O}_ X)$ the associated derived category.

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