Lemma 36.3.3 (08D3)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.3: Derived category of quasi-coherent modules
Lemma 36.3.3 (cite)

Lemma 36.3.3. Let $X$ be a scheme. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Then the map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}E$ of Derived Categories, Remark 13.34.4 is an isomorphism ¹.

Proof. Denote $\mathcal{H}^ i = H^ i(E)$ the $i$th cohomology sheaf of $E$. Let $\mathcal{B}$ be the set of affine open subsets of $X$. Then $H^ p(U, \mathcal{H}^ i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$, and all $U \in \mathcal{B}$, see Cohomology of Schemes, Lemma 30.2.2. Thus the lemma follows from Cohomology, Lemma 20.37.9. $\square$

[1] In particular, $E$ has a K-injective representative by Derived Categories, Lemma 13.34.5.

The Stacks project

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