Lemma 15.76.6. Let $R$ be a ring. Let $K \in D^-(R)$. Let $a \in \mathbf{Z}$. Assume $\mathop{\mathrm{Ext}}\nolimits ^{-a}_ R(K, M) = 0$ for any $R$-module $M$. Then there is a unique direct sum decomposition $K \cong \tau _{\leq a - 1}K \oplus \tau _{\geq a + 1}K$ and $\tau _{\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$ for some $b$.

Proof. By Lemma 15.76.5 we have a direct sum decomposition $K \cong \tau _{\leq a}K \oplus \tau _{\geq a + 1}K$ and $\tau _{\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$ for some $b$. Clearly, we must have $H^ a(K) = 0$ and we conclude that $\tau _{\leq a}K = \tau _{\leq a - 1}K$ in $D(R)$. $\square$

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