The Stacks project

Lemma 48.4.4. Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a proper morphism. If1

  1. $f$ is flat and of finite presentation, or

  2. $Y$ is Noetherian

then the equivalent conditions of Lemma 48.4.3 part (1) hold for all quasi-compact opens $V$ of $Y$.

Proof. Let $Q \in D^+_\mathit{QCoh}(\mathcal{O}_ Y)$ be supported on $Y \setminus V$. To get a contradiction, assume that $a(Q)$ is not supported on $X \setminus U$. Then we can find a perfect complex $P_ U$ on $U$ and a nonzero map $P_ U \to a(Q)|_ U$ (follows from Derived Categories of Schemes, Theorem 36.15.3). Then using Derived Categories of Schemes, Lemma 36.13.11 we may assume there is a perfect complex $P$ on $X$ and a map $P \to a(Q)$ whose restriction to $U$ is nonzero. By definition of $a$ this map is adjoint to a map $Rf_*P \to Q$.

The complex $Rf_*P$ is pseudo-coherent. In case (1) this follows from Derived Categories of Schemes, Lemma 36.30.5. In case (2) this follows from Derived Categories of Schemes, Lemmas 36.11.3 and 36.10.3. Thus we may apply Derived Categories of Schemes, Lemma 36.17.5 and get a map $I \to \mathcal{O}_ Y$ of perfect complexes whose restriction to $V$ is an isomorphism such that the composition $I \otimes ^\mathbf {L}_{\mathcal{O}_ Y} Rf_*P \to Rf_*P \to Q$ is zero. By Derived Categories of Schemes, Lemma 36.22.1 we have $I \otimes ^\mathbf {L}_{\mathcal{O}_ Y} Rf_*P = Rf_*(Lf^*I \otimes ^\mathbf {L}_{\mathcal{O}_ X} P)$. We conclude that the composition

\[ Lf^*I \otimes ^\mathbf {L}_{\mathcal{O}_ X} P \to P \to a(Q) \]

is zero. However, the restriction to $U$ is the map $P|_ U \to a(Q)|_ U$ which we assumed to be nonzero. This contradiction finishes the proof. $\square$

[1] This proof works for those morphisms of quasi-compact and quasi-separated schemes such that $Rf_*P$ is pseudo-coherent for all $P$ perfect on $X$. It follows easily from a theorem of Kiehl [Kiehl] that this holds if $f$ is proper and pseudo-coherent. This is the correct generality for this lemma and some of the other results in this chapter.

Comments (2)

Comment #4672 by Bogdan on

A typo, a complex should be changed to everywhere.

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