The Stacks project

Lemma 20.42.10. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $, $\mathcal{F}^\bullet $ be complexes of $\mathcal{O}_ X$-modules with

  1. $\mathcal{F}^ n = 0$ for $n \ll 0$,

  2. $\mathcal{E}^ n = 0$ for $n \gg 0$, and

  3. $\mathcal{E}^ n$ isomorphic to a direct summand of a finite free $\mathcal{O}_ X$-module.

Then the internal hom $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is represented by the complex $\mathcal{H}^\bullet $ with terms

\[ \mathcal{H}^ n = \bigoplus \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{F}^ p) \]

and differential as described in Section 20.38.

Proof. Choose a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ where $\mathcal{I}^\bullet $ is a bounded below complex of injectives. Note that $\mathcal{I}^\bullet $ is K-injective (Derived Categories, Lemma 13.31.4). Hence the construction in Section 20.38 shows that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is represented by the complex $(\mathcal{H}')^\bullet $ with terms

\[ (\mathcal{H}')^ n = \prod \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{I}^ p) = \bigoplus \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{I}^ p) \]

(equality because there are only finitely many nonzero terms). Note that $\mathcal{H}^\bullet $ is the total complex associated to the double complex with terms $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{F}^ p)$ and similarly for $(\mathcal{H}')^\bullet $. The natural map $(\mathcal{H}')^\bullet \to \mathcal{H}^\bullet $ comes from a map of double complexes. Thus to show this map is a quasi-isomorphism, we may use the spectral sequence of a double complex (Homology, Lemma 12.25.3)

\[ {}'E_1^{p, q} = H^ p(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{F}^\bullet )) \]

converging to $H^{p + q}(\mathcal{H}^\bullet )$ and similarly for $(\mathcal{H}')^\bullet $. To finish the proof of the lemma it suffices to show that $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ induces an isomorphism

\[ H^ p(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{F}^\bullet )) \longrightarrow H^ p(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{I}^\bullet )) \]

on cohomology sheaves whenever $\mathcal{E}$ is a direct summand of a finite free $\mathcal{O}_ X$-module. Since this is clear when $\mathcal{E}$ is finite free the result follows. $\square$


Comments (0)


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 08I5. Beware of the difference between the letter 'O' and the digit '0'.