The Stacks project

Lemma 36.13.4. Let $X$ be a quasi-compact and quasi-separated scheme. Let $E \in D^ b_\mathit{QCoh}(\mathcal{O}_ X)$. There exists an integer $n_0 > 0$ such that $\mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(\mathcal{E}, E) = 0$ for every finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ and every $n \geq n_0$.

Proof. Recall that $\mathop{\mathrm{Ext}}\nolimits ^ n_{D(\mathcal{O}_ X)}(\mathcal{E}, E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}, E[n])$. We have Mayer-Vietoris for morphisms in the derived category, see Cohomology, Lemma 20.33.3. Thus if $X = U \cup V$ and the result of the lemma holds for $E|_ U$, $E|_ V$, and $E|_{U \cap V}$ for some bound $n_0$, then the result holds for $E$ with bound $n_0 + 1$. Thus it suffices to prove the lemma when $X$ is affine, see Cohomology of Schemes, Lemma 30.4.1.

Assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. Choose a complex of $A$-modules $M^\bullet $ whose associated complex of quasi-coherent modules represents $E$, see Lemma 36.3.5. Write $\mathcal{E} = \widetilde{P}$ for some $A$-module $P$. Since $\mathcal{E}$ is finite locally free, we see that $P$ is a finite projective $A$-module. We have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}, E[n]) & = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(P, M^\bullet [n]) \\ & = \mathop{\mathrm{Hom}}\nolimits _{K(A)}(P, M^\bullet [n]) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(P, H^ n(M^\bullet )) \end{align*}

The first equality by Lemma 36.3.5, the second equality by Derived Categories, Lemma 13.19.8, and the final equality because $\mathop{\mathrm{Hom}}\nolimits _ A(P, -)$ is an exact functor. As $E$ and hence $M^\bullet $ is bounded we get zero for all sufficiently large $n$. $\square$

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