Lemma 36.4.4 (0G9Q)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.4: Total direct image
Lemma 36.4.4 (cite)

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Lemma 36.4.4. Let $X$ be a quasi-separated and quasi-compact scheme. For any object $K$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the spectral sequence

$E_2^{i, j} = H^ i(X, H^ j(K)) \Rightarrow H^{i + j}(X, K)$

of Cohomology, Example 20.29.3 is bounded and converges.

Proof. By the construction of the spectral sequence via Cohomology, Lemma 20.29.1 using the filtration given by $\tau _{\leq -p}K$ , we see that suffices to show that given $n \in \mathbf{Z}$ we have

$H^ n(X, \tau _{\leq -p}K) = 0 \text{ for } p \gg 0$

and

$H^ n(X, K) = H^ n(X, \tau _{\leq -p}K) \text{ for } p \ll 0$

The first follows from Lemma 36.3.4 applied with $F = \Gamma (X, -)$ and the bound in Cohomology of Schemes, Lemma 30.4.5. The second holds whenever $-p \leq n$ for any ringed space $(X, \mathcal{O}_ X)$ and any $K \in D(\mathcal{O}_ X)$ . $\square$

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