The Stacks project

Lemma 20.45.7. Let $X$ be a ringed space. Let $E$ be a well ordered set and let

\[ X = \bigcup \nolimits _{\alpha \in E} W_\alpha \]

be an open covering with $W_\alpha \subset W_{\alpha + 1}$ and $W_\alpha = \bigcup _{\beta < \alpha } W_\beta $ if $\alpha $ is not a successor. Let $K_\alpha $ be an object of $D(\mathcal{O}_{W_\alpha })$ with $\mathop{\mathrm{Ext}}\nolimits ^ i(K_\alpha , K_\alpha ) = 0$ for $i < 0$. Assume given isomorphisms $\rho _\beta ^\alpha : K_\alpha |_{W_\beta } \to K_\beta $ in $D(\mathcal{O}_{W_\beta })$ for all $\beta < \alpha $ with $\rho _\gamma ^\alpha = \rho _\gamma ^\beta \circ \rho ^\alpha _\beta |_{W_\gamma }$ for $\gamma < \beta < \alpha $. Then there exists an object $K$ in $D(\mathcal{O}_ X)$ and isomorphisms $K|_{W_\alpha } \to K_\alpha $ for $\alpha \in E$ compatible with the isomorphisms $\rho _\beta ^\alpha $.

Proof. In this proof $\alpha , \beta , \gamma , \ldots $ represent elements of $E$. Choose a K-injective complex $I_\alpha ^\bullet $ on $W_\alpha $ representing $K_\alpha $. For $\beta < \alpha $ denote $j_{\beta , \alpha } : W_\beta \to W_\alpha $ the inclusion morphism. Using transfinite recursion we will construct for all $\beta < \alpha $ a map of complexes

\[ \tau _{\beta , \alpha } : (j_{\beta , \alpha })_!I_\beta ^\bullet \longrightarrow I_\alpha ^\bullet \]

representing the adjoint to the inverse of the isomorphism $\rho ^\alpha _\beta : K_\alpha |_{W_\beta } \to K_\beta $. Moreover, we will do this in such that for $\gamma < \beta < \alpha $ we have

\[ \tau _{\gamma , \alpha } = \tau _{\beta , \alpha } \circ (j_{\beta , \alpha })_!\tau _{\gamma , \beta } \]

as maps of complexes. Namely, suppose already given $\tau _{\gamma , \beta }$ composing correctly for all $\gamma < \beta < \alpha $. If $\alpha = \alpha ' + 1$ is a successor, then we choose any map of complexes

\[ (j_{\alpha ', \alpha })_!I_{\alpha '}^\bullet \to I_\alpha ^\bullet \]

which is adjoint to the inverse of the isomorphism $\rho ^\alpha _{\alpha '} : K_\alpha |_{W_{\alpha '}} \to K_{\alpha '}$ (possible because $I_\alpha ^\bullet $ is K-injective) and for any $\beta < \alpha '$ we set

\[ \tau _{\beta , \alpha } = \tau _{\alpha ', \alpha } \circ (j_{\alpha ', \alpha })_!\tau _{\beta , \alpha '} \]

If $\alpha $ is not a successor, then we can consider the complex on $W_\alpha $ given by

\[ C^\bullet = \mathop{\mathrm{colim}}\nolimits _{\beta < \alpha } (j_{\beta , \alpha })_!I_\beta ^\bullet \]

(termwise colimit) where the transition maps of the sequence are given by the maps $\tau _{\beta ', \beta }$ for $\beta ' < \beta < \alpha $. We claim that $C^\bullet $ represents $K_\alpha $. Namely, for $\beta < \alpha $ the restriction of the coprojection $(j_{\beta , \alpha })_!I_\beta ^\bullet \to C^\bullet $ gives a map

\[ \sigma _\beta : I_\beta ^\bullet \longrightarrow C^\bullet |_{W_\beta } \]

which is a quasi-isomorphism: if $x \in W_\beta $ then looking at stalks we get

\[ (C^\bullet )_ x = \mathop{\mathrm{colim}}\nolimits _{\beta ' < \alpha } \left((j_{\beta ', \alpha })_!I_{\beta '}^\bullet \right)_ x = \mathop{\mathrm{colim}}\nolimits _{\beta \leq \beta ' < \alpha } (I_{\beta '}^\bullet )_ x \longleftarrow (I_\beta ^\bullet )_ x \]

which is a quasi-isomorphism. Here we used that taking stalks commutes with colimits, that filtered colimits are exact, and that the maps $(I_\beta ^\bullet )_ x \to (I_{\beta '}^\bullet )_ x$ are quasi-isomorphisms for $\beta \leq \beta ' < \alpha $. Hence $(C^\bullet , \sigma _\beta ^{-1})$ is a solution to the system $(\{ K_\beta \} _{\beta < \alpha }, \{ \rho ^\beta _{\beta '}\} _{\beta ' < \beta < \alpha })$. Since $(K_\alpha , \rho ^\alpha _\beta )$ is another solution we obtain a unique isomorphism $\sigma : K_\alpha \to C^\bullet $ in $D(\mathcal{O}_{W_\alpha })$ compatible with all our maps, see Lemma 20.45.6 (this is where we use the vanishing of negative ext groups). Choose a morphism $\tau : C^\bullet \to I_\alpha ^\bullet $ of complexes representing $\sigma $. Then we set

\[ \tau _{\beta , \alpha } = \tau |_{W_\beta } \circ \sigma _\beta \]

to get the desired maps. Finally, we take $K$ to be the object of the derived category represented by the complex

\[ K^\bullet = \mathop{\mathrm{colim}}\nolimits _{\alpha \in E} (W_\alpha \to X)_!I_\alpha ^\bullet \]

where the transition maps are given by our carefully constructed maps $\tau _{\beta , \alpha }$ for $\beta < \alpha $. Arguing exactly as above we see that for all $\alpha $ the restriction of the coprojection determines an isomorphism

\[ K|_{W_\alpha } \longrightarrow K_\alpha \]

compatible with the given maps $\rho ^\alpha _\beta $. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 20.45: Glueing complexes

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 0D6B. Beware of the difference between the letter 'O' and the digit '0'.