The Stacks project

Lemma 13.27.2. Let $\mathcal{A}$ be an abelian category. Let $X^\bullet , Y^\bullet \in \mathop{\mathrm{Ob}}\nolimits (K(\mathcal{A}))$.

  1. Let $Y^\bullet \to I^\bullet $ be an injective resolution (Definition 13.18.1). Then

    \[ \mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(X^\bullet , Y^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(X^\bullet , I^\bullet [i]). \]

  2. Let $P^\bullet \to X^\bullet $ be a projective resolution (Definition 13.19.1). Then

    \[ \mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(X^\bullet , Y^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(P^\bullet [-i], Y^\bullet ). \]

Proof. Follows immediately from Lemma 13.18.8 and Lemma 13.19.8. $\square$

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