Lemma 75.22.2. Let S be a scheme. Let B be a Noetherian algebraic space over S. Let f : X \to B be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let E \in D(\mathcal{O}_ X) be perfect. Let \mathcal{G}^\bullet be a bounded complex of coherent \mathcal{O}_ X-modules flat over B with support proper over B. Then K = Rf_*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet ) is a perfect object of D(\mathcal{O}_ B).
Proof. The object K is perfect by Lemma 75.22.1. We check the lemma applies: Locally E is isomorphic to a finite complex of finite free \mathcal{O}_ X-modules. Hence locally E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet is isomorphic to a finite complex whose terms are of the form
\bigoplus \nolimits _{i = a, \ldots , b} (\mathcal{G}^ i)^{\oplus r_ i}
for some integers a, b, r_ a, \ldots , r_ b. This immediately implies the cohomology sheaves H^ i(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}) are coherent. The hypothesis on the tor dimension also follows as \mathcal{G}^ i is flat over f^{-1}\mathcal{O}_ Y. \square
Comments (0)