Lemma 37.69.6. Let $A$ be a ring. Let $X$ be a scheme 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 37.69.5, $K$ is pseudo-coherent relative to $A$. By Lemma 37.59.18, $K$ is pseudo-coherent in $D( \mathcal{O}_ X)$. By Derived Categories of Schemes, Proposition 36.40.6 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 Derived Categories of Schemes, Lemma 36.35.13, 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 Schemes, Theorem 36.15.3. We have

The first equality uses that $Li^*$ preserves perfect objects and duals and Cohomology, Lemma 20.50.5; we omit some details. The second equality follows from Derived Categories of Schemes, Lemma 36.22.5 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 Schemes, Remark 36.16.4. Hence Derived Categories of Schemes, Proposition 36.40.6 now implies that $Li^*(K)$ is bounded below and we win. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)