Lemma 15.66.5. Let $R$ be a ring. Let $(K^\bullet , L^\bullet , M^\bullet , f, g, h)$ be a distinguished triangle in $D(R)$. Let $a, b \in \mathbf{Z}$.

If $K^\bullet $ has tor-amplitude in $[a + 1, b + 1]$ and $L^\bullet $ has tor-amplitude in $[a, b]$ then $M^\bullet $ has tor-amplitude in $[a, b]$.

If $K^\bullet , M^\bullet $ have tor-amplitude in $[a, b]$, then $L^\bullet $ has tor-amplitude in $[a, b]$.

If $L^\bullet $ has tor-amplitude in $[a + 1, b + 1]$ and $M^\bullet $ has tor-amplitude in $[a, b]$, then $K^\bullet $ has tor-amplitude in $[a + 1, b + 1]$.

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