The Stacks project

Lemma 36.18.1. Let $X$ be a scheme. 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) and 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09M3. Beware of the difference between the letter 'O' and the digit '0'.