Lemma 13.27.3. Let $\mathcal{A}$ be an abelian category.

Let $X$, $Y$ be objects of $D(\mathcal{A})$. Given $a, b \in \mathbf{Z}$ such that $H^ i(X) = 0$ for $i > a$ and $H^ j(Y) = 0$ for $j < b$, we have $\mathop{\mathrm{Ext}}\nolimits ^ n_\mathcal {A}(X, Y) = 0$ for $n < b - a$ and

\[ \mathop{\mathrm{Ext}}\nolimits ^{b - a}_\mathcal {A}(X, Y) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(H^ a(X), H^ b(Y)) \]Let $A, B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. For $i < 0$ we have $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(B, A) = 0$. We have $\mathop{\mathrm{Ext}}\nolimits ^0_\mathcal {A}(B, A) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(B, A)$.

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