The Stacks project

Lemma 20.42.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L, M$ be objects of $D(\mathcal{O}_ X)$. With the construction as described above there is a canonical isomorphism

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

in $D(\mathcal{O}_ X)$ functorial in $K, L, M$ which recovers (20.42.0.1) by taking $H^0(X, -)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $M$ and a K-flat complex of $\mathcal{O}_ X$-modules $\mathcal{L}^\bullet $ representing $L$. Let $\mathcal{K}^\bullet $ be any complex of $\mathcal{O}_ X$-modules representing $K$. Then we have

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

by Lemma 20.41.1. Note that the left hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M))$ (use Lemma 20.41.8) and that the right hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} L, M)$. This proves the displayed formula of the lemma. Taking global sections and using Lemma 20.42.1 we obtain (20.42.0.1). $\square$

Comments (2)

Comment #6719 by Bach on

There is suddenly a complex H in the proof of 08DJ.

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Your email address will not be published.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 08DJ. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 20.42.2 as pdf