Lemma 75.17.1 (09MA)—The Stacks project

Lemma 75.17.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $K^\bullet$ be a complex of $\mathcal{O}_ X$ -modules whose cohomology sheaves are quasi-coherent. Let $(E, d) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(K^\bullet , K^\bullet )$ be the endomorphism differential graded algebra. Then the functor

$- \otimes _ E^\mathbf {L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_ X)$

of Differential Graded Algebra, Lemma 22.35.3 has image contained in $D_\mathit{QCoh}(\mathcal{O}_ X)$ .

Proof. Let $P$ be a differential graded $E$ -module with property $P$ . Let $F_\bullet$ be a filtration on $P$ as in Differential Graded Algebra, Section 22.20. Then we have

$P \otimes _ E K^\bullet = \text{hocolim}\ F_ iP \otimes _ E K^\bullet$

Each of the $F_ iP$ has a finite filtration whose graded pieces are direct sums of $E[k]$ . The result follows easily. $\square$

The Stacks project

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