Definition 13.27.1. Let $\mathcal{A}$ be an abelian category. Let $i \in \mathbf{Z}$. Let $X, Y$ be objects of $D(\mathcal{A})$. The $i$th extension group of $X$ by $Y$ is the group
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(X, Y) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(X, Y[i]) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(X[-i], Y). \]
If $A, B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ we set $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(A, B) = \text{Ext}^ i_\mathcal {A}(A[0], B[0])$.
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