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 $a, b \in \mathbf{Z}$.

1. 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 $[a, b]$.

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

3. 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 $[a + 1, b + 1]$.

Proof. Omitted. Hint: This just follows from the long exact cohomology sequence associated to a distinguished triangle and the fact that $- \otimes _ R^{\mathbf{L}} M$ preserves distinguished triangles. The easiest one to prove is (2) and the others follow from it by translation. $\square$

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