The Stacks project

Lemma 20.30.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_ X$-modules. There exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{G}^\bullet $ where $\mathcal{F}^\bullet $ be a bounded below complex of flasque $\mathcal{O}_ X$-modules and for all $x \in X$ the map $\mathcal{F}^\bullet _ x \to \mathcal{G}^\bullet _ x$ is a homotopy equivalence in the category of complexes of $\mathcal{O}_{X, x}$-modules.

Proof. Let $\mathcal{A}$ be the category of complexes of $\mathcal{O}_ X$-modules and let $\mathcal{B}$ be the category of complexes of $\mathcal{O}_ X$-modules. Then we can apply the discussion above to the adjoint functors $f^*$ and $f_*$ between $\mathcal{A}$ and $\mathcal{B}$. Arguing exactly as in the proof of Lemma 20.30.1 we get a resolution

\[ 0 \to \mathcal{F}^\bullet \to f_*f^*\mathcal{F}^\bullet \to f_*f^*f_*f^*\mathcal{F}^\bullet \to f_*f^*f_*f^*f_*f^*\mathcal{F}^\bullet \to \ldots \]

in the abelian category $\mathcal{A}$ such that each term of each $f_*f^*\ldots f_*f^*\mathcal{F}^\bullet $ is a flasque $\mathcal{O}_ X$-module and such that for all $x \in X$ the map

\[ \mathcal{F}^\bullet _ x[0] \to \Big( (f_*f^*\mathcal{F}^\bullet )_ x \to (f_*f^*f_*f^*\mathcal{F}^\bullet )_ x \to (f_*f^*f_*f^*f_*f^*\mathcal{F}^\bullet )_ x \to \ldots \Big) \]

is a homotopy equivalence in the category of complexes of complexes of $\mathcal{O}_{X, x}$-modules. Since a complex of complexes is the same thing as a double complex, we can consider the induced map

\[ \mathcal{F}^\bullet \to \mathcal{G}^\bullet = \text{Tot}( f_*f^*\mathcal{F}^\bullet \to f_*f^*f_*f^*\mathcal{F}^\bullet \to f_*f^*f_*f^*f_*f^*\mathcal{F}^\bullet \to \ldots ) \]

Since the complex $\mathcal{F}^\bullet $ is bounded below, the same is true for $\mathcal{G}^\bullet $ and in fact each term of $\mathcal{G}^\bullet $ is a finite direct sum of terms of the complexes $f_*f^*\ldots f_*f^*\mathcal{F}^\bullet $ and hence is flasque. The final assertion of the lemma now follows from Homology, Lemma 12.25.5. Since this in particular shows that $\mathcal{F}^\bullet \to \mathcal{G}^\bullet $ is a quasi-isomorphism, the proof is complete. $\square$


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Your email address will not be published.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