Lemma 15.76.1. Let $R$ be a ring. Let $K$ and $L$ be objects of $D(R)$. Assume $L$ has projective-amplitude in $[a, b]$, for example if $L$ is perfect of tor-amplitude in $[a, b]$.

1. If $H^ i(K) = 0$ for $i \geq a$, then $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(L, K) = 0$.

2. If $H^ i(K) = 0$ for $i \geq a + 1$, then given any distinguished triangle $K \to M \to L \to K[1]$ there is an isomorphism $M \cong K \oplus L$ in $D(R)$ compatible with the maps in the distinguished triangle.

3. If $H^ i(K) = 0$ for $i \geq a$, then the isomorphism in (2) exists and is unique.

Proof. The assumption that $L$ has projective-amplitude in $[a, b]$ means we can represent $L$ by a complex $L^\bullet$ of projective $R$-modules with $L^ i = 0$ for $i \not\in [a, b]$, see Definition 15.68.1. If $L$ is perfect of tor-amplitude in $[a, b]$, then we can represent $L$ by a complex $L^\bullet$ of finite projective $R$-modules with $L^ i = 0$ for $i \not\in [a, b]$, see Lemma 15.74.2. If $H^ i(K) = 0$ for $i \geq a$, then $K$ is quasi-isomorphic to $\tau _{\leq a - 1}K$. Hence we can represent $K$ by a complex $K^\bullet$ of $R$-modules with $K^ i = 0$ for $i \geq a$. Then we obtain

$\mathop{\mathrm{Hom}}\nolimits _{D(R)}(L, K) = \mathop{\mathrm{Hom}}\nolimits _{K(R)}(L^\bullet , K^\bullet ) = 0$

by Derived Categories, Lemma 13.19.8. This proves (1). Under the hypotheses of (2) we see that $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(L, K[1]) = 0$ by (1), hence the distinguished triangle is split by Derived Categories, Lemma 13.4.11. The uniqueness of (3) follows from (1). $\square$

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