Lemma 84.5.3. With notation and hypotheses as in Lemma 84.5.2. For $K \in D(\mathcal{C}_{total})$ we have $(g'_ n)^{-1}Rh_{total, *}K = Rh_{n, *}g_ n^{-1}K$.

**Proof.**
Let $\mathcal{I}^\bullet $ be a K-injective complex on $\mathcal{C}_{total}$ representing $K$. Then $g_ n^{-1}K$ is represented by $g_ n^{-1}\mathcal{I}^\bullet = \mathcal{I}_ n^\bullet $ which is K-injective by Lemma 84.3.6. We have $(g'_ n)^{-1}h_{total, *}\mathcal{I}^\bullet = h_{n, *}g_ n^{-1}\mathcal{I}_ n^\bullet $ by Lemma 84.5.2 which gives the desired equality.
$\square$

## 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Your email address will not be published.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)