Lemma 20.38.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given $K, L$ in $D(\mathcal{O}_ X)$ there is a canonical morphism

in $D(\mathcal{O}_ X)$ functorial in both $K$ and $L$.

Lemma 20.38.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given $K, L$ in $D(\mathcal{O}_ X)$ there is a canonical morphism

\[ K \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

in $D(\mathcal{O}_ X)$ functorial in both $K$ and $L$.

**Proof.**
Choose a K-flat complex $\mathcal{K}^\bullet $ representing $K$ and any complex of $\mathcal{O}_ X$-modules $\mathcal{L}^\bullet $ representing $L$. Choose a K-injective complex $\mathcal{J}^\bullet $ and a quasi-isomorphism $\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ) \to \mathcal{J}^\bullet $. Then we use

\[ \mathcal{K}^\bullet \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet )) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{J}^\bullet ) \]

where the first map comes from Lemma 20.37.4. $\square$

Your email address will not be published.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)