Lemma 15.103.6. Let $R$ be a ring and $f \in R$. Let $M \in D(R)$ and let $C$ be the cone of $f : M \to M$. If $H^ i(M)_ f = 0$ for $i < 0$ and $H^ i(C) = 0$ for $i < -1$, then $H^ i(M) = 0$ for $i < 0$.
Proof. Denote $M_ f = M \otimes _ R^\mathbf {L} R_ f$ and choose a distinguished triangle $F \to M \to M_ f$. By assumption $H^ i(M_ f) = H^ i(M)_ f = 0$ for $i < 0$. Thus it suffices to show that $H^ i(F) = 0$ for $i < 0$. Note that $H^ i(F)$ is $f$-power torsion for all $i \in \mathbf{Z}$. On the other hand, since $f : M_ f \to M_ f$ is an isomorphism, we see that $C$ is isomorphic to the cone of $f : F \to F$ (use Derived Categories, Proposition 13.4.23). Now, if $H^ i(F) \not= 0$, then the kernel of $f : H^ i(F) \to H^ i(F)$ is nonzero, which implies that $H^{i - 1}(C)$ is nonzero. Our assumption implies this cannot happen if $i - 1 < -1$ which finishes the proof. $\square$
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