Lemma 48.3.10. Let $f : X \to Y$ be a morphism of quasi-separated and quasi-compact schemes. For all $L \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ (48.3.5.1) induces an isomorphism $R\mathop{\mathrm{Hom}}\nolimits _ X(L, a(K)) \to R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*L, K)$ of global derived homs.

**Proof.**
By the construction in Cohomology, Section 20.44 we have

and

Thus the lemma is a consequence of Lemma 48.3.6. Namely, a map $E \to E'$ in $D(\mathcal{O}_ Y)$ which induces an isomorphism $DQ_ Y(E) \to DQ_ Y(E')$ induces a quasi-isomorphism $R\Gamma (Y, E) \to R\Gamma (Y, E')$. Indeed we have $H^ i(Y, E) = \mathop{\mathrm{Ext}}\nolimits ^ i_ Y(\mathcal{O}_ Y, E) = \mathop{\mathrm{Hom}}\nolimits (\mathcal{O}_ Y[-i], E) = \mathop{\mathrm{Hom}}\nolimits (\mathcal{O}_ Y[-i], DQ_ Y(E))$ because $\mathcal{O}_ Y[-i]$ is in $D_\mathit{QCoh}(\mathcal{O}_ Y)$ and $DQ_ Y$ is the right adjoint to the inclusion functor $D_\mathit{QCoh}(\mathcal{O}_ Y) \to D(\mathcal{O}_ Y)$. $\square$

## 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Your email address will not be published.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 (2)

Comment #4925 by awllower on

Comment #5193 by Johan on