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The Stacks project

Lemma 36.27.1. Let S be a Noetherian scheme. Let f : X \to S be a morphism of schemes which is locally of finite type. Let E \in D(\mathcal{O}_ X) such that

  1. E \in D^ b_{\textit{Coh}}(\mathcal{O}_ X),

  2. the support of H^ i(E) is proper over S for all i, and

  3. E has finite tor dimension as an object of D(f^{-1}\mathcal{O}_ S).

Then Rf_*E is a perfect object of D(\mathcal{O}_ S).

Proof. By Lemma 36.11.3 we see that Rf_*E is an object of D^ b_{\textit{Coh}}(\mathcal{O}_ S). Hence Rf_*E is pseudo-coherent (Lemma 36.10.3). Hence it suffices to show that Rf_*E has finite tor dimension, see Cohomology, Lemma 20.49.5. By Lemma 36.10.6 it suffices to check that Rf_*(E) \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{F} has universally bounded cohomology for all quasi-coherent sheaves \mathcal{F} on S. Bounded from above is clear as Rf_*(E) is bounded from above. Let T \subset X be the union of the supports of H^ i(E) for all i. Then T is proper over S by assumptions (1) and (2), see Cohomology of Schemes, Lemma 30.26.6. In particular there exists a quasi-compact open X' \subset X containing T. Setting f' = f|_{X'} we have Rf_*(E) = Rf'_*(E|_{X'}) because E restricts to zero on X \setminus T. Thus we may replace X by X' and assume f is quasi-compact. Moreover, f is quasi-separated by Morphisms, Lemma 29.15.7. Now

Rf_*(E) \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{F} = Rf_*\left(E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F}\right) = Rf_*\left(E \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} f^{-1}\mathcal{F}\right)

by Lemma 36.22.1 and Cohomology, Lemma 20.27.4. By assumption (3) the complex E \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} f^{-1}\mathcal{F} has cohomology sheaves in a given finite range, say [a, b]. Then Rf_* of it has cohomology in the range [a, \infty ) and we win. \square


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