The Stacks project

Proof. Part (2) follows from (1) as $\mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(K, L) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L[n])$. We prove (1). Since $K$ is perfect we have

where $K^\vee $ is the “dual” perfect complex to $K$, see Cohomology on Sites, Lemma 21.48.4. Note that $P = K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ is in $D_\mathit{QCoh}(X)$ by Lemmas 75.5.6 and 75.13.6 (to see that a perfect complex has quasi-coherent cohomology sheaves). Say $K^\vee $ has tor amplitude in $[a, b]$. Then the spectral sequence

shows that $H^ j(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L)$ is zero if $H^ q(L) = 0$ for $q \in [j - b, j - a]$. Let $N$ be the integer $\max (d_ p + p)$ of Cohomology of Spaces, Lemma 69.7.3. Then $H^0(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L)$ vanishes if the cohomology sheaves

are zero. Namely, by the lemma cited and Lemma 75.5.8, we have

and by the vanishing of cohomology sheaves, this is equal to $H^0(X, \tau _{\geq 1}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L))$ which is zero by Derived Categories, Lemma 13.16.1. It follows that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L)$ is zero if $H^ i(L) = 0$ for $i \in [-b - N, -a]$. $\square$