The Stacks project

Lemma 20.39.7. 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. By transfinite induction, 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 20.39.6 (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 $. $\square$


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