Lemma 13.15.2. In Situation 13.15.1.

Let $X$ be an object of $K^{+}(\mathcal{A})$. The right derived functor of $K(\mathcal{A}) \to D(\mathcal{B})$ is defined at $X$ if and only if the right derived functor of $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is defined at $X$. Moreover, the values are canonically isomorphic.

Let $X$ be an object of $K^{+}(\mathcal{A})$. Then $X$ computes the right derived functor of $K(\mathcal{A}) \to D(\mathcal{B})$ if and only if $X$ computes the right derived functor of $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.

Let $X$ be an object of $K^{-}(\mathcal{A})$. The left derived functor of $K(\mathcal{A}) \to D(\mathcal{B})$ is defined at $X$ if and only if the left derived functor of $K^{-}(\mathcal{A}) \to D^{-}(\mathcal{B})$ is defined at $X$. Moreover, the values are canonically isomorphic.

Let $X$ be an object of $K^{-}(\mathcal{A})$. Then $X$ computes the left derived functor of $K(\mathcal{A}) \to D(\mathcal{B})$ if and only if $X$ computes the left derived functor of $K^{-}(\mathcal{A}) \to D^{-}(\mathcal{B})$.

## Comments (1)

Comment #8395 by ElĂas Guisado on

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