Lemma 13.27.7. 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

This is possible by Lemma 13.27.5. Set $C = \mathop{\mathrm{Ker}}(Z_{p - 1} \to Z_ p) = \mathop{\mathrm{Im}}(Z_ p \to Z_{p - 1})$. Then $\delta (E)$ is the composition of $\delta (E')$ and $\delta (E'')$ where

and

Since $\delta (E') \in \mathop{\mathrm{Ext}}\nolimits ^ p_\mathcal {A}(B, C) = 0$ we conclude. $\square$

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