The Stacks project

Lemma 74.17.2. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K$ be a perfect object of $D(\mathcal{O}_ X)$. Then

  1. there exist integers $a \leq b$ such that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L) = 0$ for $L \in D_\mathit{QCoh}(\mathcal{O}_ X)$ with $H^ i(L) = 0$ for $i \in [a, b]$, and

  2. if $L$ is bounded, then $\mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(K, L)$ is zero for all but finitely many $n$.

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

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{D(\mathcal{O}_ X)}(K, L) = H^ i(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

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 74.5.6 and 74.13.6 (to see that a perfect complex has quasi-coherent cohomology sheaves). Say $K^\vee $ has tor amplitude in $[a, b]$. Then the spectral sequence

\[ E_1^{p, q} = H^ p(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} H^ q(L)) \Rightarrow H^{p + q}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

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 68.7.3. Then $H^0(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L)$ vanishes if the cohomology sheaves

\[ H^{-N}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L), \ H^{-N + 1}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L), \ \ldots , \ H^0(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

are zero. Namely, by the lemma cited and Lemma 74.5.8, we have

\[ H^0(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L) = H^0(X, \tau _{\geq -N}(K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L)) \]

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 $i \in [-b - N, -a]$. $\square$


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