Lemma 22.35.3. The functor $K(\text{Mod}_{(E, \text{d})}) \to K(\mathcal{O})$ of Lemma 22.35.2 has a left derived version defined on all of $D(E, \text{d})$. We denote it $- \otimes _ E^\mathbf {L} K^\bullet : D(E, \text{d}) \to D(\mathcal{O})$.

**Proof.**
We will use Derived Categories, Lemma 13.14.15 to prove this. As our collection $\mathcal{P}$ of objects we will use the objects with property (P). Property (1) was shown in Lemma 22.20.4. Property (2) holds because if $s : P \to P'$ is a quasi-isomorphism of modules with property (P), then $s$ is a homotopy equivalence by Lemma 22.22.3.
$\square$

## 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Your email address will not be published.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.

## Comments (0)