Lemma 15.57.4. Let $R$ be a ring. Let $K^\bullet $ be a K-flat complex. Then the functor

transforms quasi-isomorphisms into quasi-isomorphisms.

Lemma 15.57.4. Let $R$ be a ring. Let $K^\bullet $ be a K-flat complex. Then the functor

\[ K(\text{Mod}_ R) \longrightarrow K(\text{Mod}_ R), \quad L^\bullet \longmapsto \text{Tot}(L^\bullet \otimes _ R K^\bullet ) \]

transforms quasi-isomorphisms into quasi-isomorphisms.

**Proof.**
Follows from Lemma 15.57.2 and the fact that quasi-isomorphisms in $K(\text{Mod}_ R)$ and $K(\text{Mod}_ A)$ are characterized by having acyclic cones.
$\square$

Your email address will not be published.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)

There are also: