Lemma 13.27.8. Let $\mathcal{A}$ be an abelian category and let $p \geq 0$. If $\mathop{\mathrm{Ext}}\nolimits ^ p_\mathcal {A}(B, A) = 0$ for any pair of objects $A$, $B$ of $\mathcal{A}$, then $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {A}(B, A) = 0$ for $i \geq p$ and any pair of objects $A$, $B$ of $\mathcal{A}$.
Proof. For $i > p$ write any class $\xi $ as $\delta (E)$ where $E$ is a Yoneda extension
\[ E : 0 \to A \to Z_{i - 1} \to Z_{i - 2} \to \ldots \to Z_0 \to B \to 0 \]
This is possible by Lemma 13.27.5. Set $C = \mathop{\mathrm{Ker}}(Z_{p - 1} \to Z_{p - 2}) = \mathop{\mathrm{Im}}(Z_ p \to Z_{p - 1})$. Then $\delta (E)$ is the composition of $\delta (E')$ and $\delta (E'')$ where
\[ E' : 0 \to C \to Z_{p - 1} \to \ldots \to Z_0 \to B \to 0 \]
and
\[ E'' : 0 \to A \to Z_{i - 1} \to Z_{i - 2} \to \ldots \to Z_ p \to C \to 0 \]
Since $\delta (E') \in \mathop{\mathrm{Ext}}\nolimits ^ p_\mathcal {A}(B, C) = 0$ we conclude. $\square$
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