The Stacks project

Lemma 20.42.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given $K, L, M$ in $D(\mathcal{O}_ X)$ there is a canonical morphism

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

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

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

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

where the first map is the map from Lemma 20.41.3. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 20.42: Internal hom in the derived category

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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BYS. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 20.42.6 as pdf