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The Stacks project

Lemma 52.6.8. Let (\mathcal{C}, \mathcal{O}) be a ringed on a site. Let f_1, \ldots , f_ r be global sections of \mathcal{O}. Let \mathcal{I} \subset \mathcal{O} be the ideal sheaf generated by f_1, \ldots , f_ r. Then the inclusion functor D_{comp}(\mathcal{O}) \to D(\mathcal{O}) has a left adjoint, i.e., given any object K of D(\mathcal{O}) there exists a map K \to K^\wedge with K^\wedge in D_{comp}(\mathcal{O}) such that the map

\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K^\wedge , E) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K, E)

is bijective whenever E is in D_{comp}(\mathcal{O}). In fact we have

K^\wedge = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{O} \to \prod \nolimits _{i_0} \mathcal{O}_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_ r}, K)

functorially in K.

Proof. Define K^\wedge by the last displayed formula of the lemma. There is a map of complexes

(\mathcal{O} \to \prod \nolimits _{i_0} \mathcal{O}_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{f_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{f_1\ldots f_ r}) \longrightarrow \mathcal{O}

which induces a map K \to K^\wedge . It suffices to prove that K^\wedge is derived complete and that K \to K^\wedge is an isomorphism if K is derived complete.

Let f be a global section of \mathcal{O}. By Lemma 52.6.1 the object R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K^\wedge ) is equal to

R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}( (\mathcal{O}_ f \to \prod \nolimits _{i_0} \mathcal{O}_{ff_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{ff_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{ff_1\ldots f_ r}), K)

If f = f_ i for some i, then f_1, \ldots , f_ r generate the unit ideal in \mathcal{O}_ f, hence the extended alternating Čech complex

\mathcal{O}_ f \to \prod \nolimits _{i_0} \mathcal{O}_{ff_{i_0}} \to \prod \nolimits _{i_0 < i_1} \mathcal{O}_{ff_{i_0}f_{i_1}} \to \ldots \to \mathcal{O}_{ff_1\ldots f_ r}

is zero (even homotopic to zero). In this way we see that K^\wedge is derived complete.

If K is derived complete, then R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K) is zero for all f = f_{i_0} \ldots f_{i_ p}, p \geq 0. Thus K \to K^\wedge is an isomorphism in D(\mathcal{O}). \square


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