The Stacks project

Lemma 76.52.14. Let $A$ be a ring. Let $X$ be an algebraic space separated, of finite presentation, and flat over $A$. Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$. If $R \Gamma (X, E \otimes ^\mathbf {L} K)$ is perfect in $D(A)$ for every perfect $E \in D(\mathcal{O}_ X)$, then $K$ is $\mathop{\mathrm{Spec}}(A)$-perfect.

Proof. By Lemma 76.51.5, $K$ is pseudo-coherent relative to $A$. By Lemma 76.45.4, $K$ is pseudo-coherent in $D(\mathcal{O}_ X)$. By Derived Categories of Spaces, Proposition 75.29.4 we see that $K$ is in $D^-(\mathcal{O}_ X)$. Let $\mathfrak {p}$ be a prime ideal of $A$ and denote $i : Y \to X$ the inclusion of the scheme theoretic fibre over $\mathfrak {p}$, i.e., $Y$ is a scheme over $\kappa (\mathfrak p)$. By Lemma 76.52.13, we will be done if we can show $Li^*(K)$ is bounded below. Let $G \in D_{perf} (\mathcal{O}_ X)$ be a perfect complex which generates $D_\mathit{QCoh}(\mathcal{O}_ X)$, see Derived Categories of Spaces, Theorem 75.15.4. We have

\begin{align*} R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ Y}(Li^*(G), Li^*(K)) & = R\Gamma (Y, Li^*(G ^\vee \otimes ^\mathbf {L} K)) \\ & = R\Gamma (X, G^\vee \otimes ^{\mathbf{L}} K) \otimes ^\mathbf {L}_ A \kappa (\mathfrak {p}) \end{align*}

The first equality uses that $Li^*$ preserves perfect objects and duals and Cohomology on Sites, Lemma 21.48.4; we omit some details. The second equality follows from Derived Categories of Spaces, Lemma 75.20.4 as $X$ is flat over $A$. It follows from our hypothesis that this is a perfect object of $D(\kappa (\mathfrak {p}))$. The object $Li^*(G) \in D_{perf}(\mathcal{O}_ Y)$ generates $D_\mathit{QCoh}(\mathcal{O}_ Y)$ by Derived Categories of Spaces, Remark 75.15.5. Hence Derived Categories of Spaces, Proposition 75.29.4 now implies that $Li^*(K)$ is bounded below and we win. $\square$

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