The Stacks project

Lemma 15.79.4. Let $R$ be a Noetherian regular ring of dimension $1 \leq d < \infty $. Let $K \in D(R)$ be perfect and let $k \in \mathbf{Z}$ such that $H^ i(K) = 0$ for $i = k - d + 2, \ldots , k$ (empty condition if $d = 1$). Then $K = \tau _{\leq k - d + 1}K \oplus \tau _{\geq k + 1}K$.

Proof. The vanishing of cohomology shows that we have a distinguished triangle

\[ \tau _{\leq k - d + 1}K \to K \to \tau _{\geq k + 1}K \to (\tau _{\leq k - d + 1}K)[1] \]

By Derived Categories, Lemma 13.4.11 it suffices to show that the third arrow is zero. Thus it suffices to show that $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(\tau _{\geq k + 1}K, (\tau _{\leq k - d + 1}K)[1]) = 0$ which follows from Lemma 15.79.3. $\square$

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