The Stacks project

Lemma 20.53.7. In Situation 20.53.2 there is an additive functor1 $L\eta _\mathcal {I} : D(\mathcal{O}_ X) \to D(\mathcal{O}_ X)$ such that if $M$ in $D(\mathcal{O}_ X)$ is represented by a complex $\mathcal{F}^\bullet $ of $\mathcal{I}$-torsion free $\mathcal{O}_ X$-modules, then $L\eta _\mathcal {I}M = \eta _\mathcal {I}\mathcal{F}^\bullet $. Similarly for morphisms.

Proof. Denote $\mathcal{T} \subset \textit{Mod}(\mathcal{O}_ X)$ the full subcategory of $\mathcal{I}$-torsion free $\mathcal{O}_ X$-modules. We have a corresponding inclusion

\[ K(\mathcal{T}) \quad \subset \quad K(\textit{Mod}(\mathcal{O}_ X)) = K(\mathcal{O}_ X) \]

of $K(\mathcal{T})$ as a full triangulated subcategory of $K(\mathcal{O}_ X)$. Let $S \subset \text{Arrows}(K(\mathcal{T}))$ be the quasi-isomorphisms. We will apply Derived Categories, Lemma 13.5.7 to show that the map

\[ S^{-1}K(\mathcal{T}) \longrightarrow D(\mathcal{O}_ X) \]

is an equivalence of triangulated categories. The lemma shows that it suffices to prove: given a complex $\mathcal{G}^\bullet $ of $\mathcal{O}_ X$-modules, there exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{G}^\bullet $ with $\mathcal{F}^\bullet $ a complex of $\mathcal{I}$-torsion free $\mathcal{O}_ X$-modules. By Lemma 20.26.12 we can find a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{G}^\bullet $ such that the complex $\mathcal{F}^\bullet $ is K-flat (we won't use this) and consists of flat $\mathcal{O}_ X$-modules $\mathcal{F}^ i$. By the third characterization of Lemma 20.53.3 we see that a flat $\mathcal{O}_ X$-module is an $\mathcal{I}$-torsion free $\mathcal{O}_ X$-module and we are done.

With these preliminaries out of the way we can define $L\eta _ f$. Namely, by the discussion following Lemma 20.53.3 this section we have already a well defined functor

\[ K(\mathcal{T}) \xrightarrow {\eta _ f} K(\mathcal{T}) \to K(\mathcal{O}_ X) \to D(\mathcal{O}_ X) \]

which according to Lemma 20.53.6 sends quasi-isomorphisms to quasi-isomorphisms. Hence this functor factors over $S^{-1}K(\mathcal{T}) = D(\mathcal{O}_ X)$ by Categories, Lemma 4.27.8. $\square$

[1] Beware that this functor isn't exact, i.e., does not transform distinguished triangles into distinguished triangles.

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Your email address will not be published.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 0F8Q. Beware of the difference between the letter 'O' and the digit '0'.