Lemma 52.6.9. 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. Let K \in D(\mathcal{O}). The derived completion K^\wedge of Lemma 52.6.8 is given by the formula
K^\wedge = R\mathop{\mathrm{lim}}\nolimits K \otimes ^\mathbf {L}_\mathcal {O} K_ n
where K_ n = K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) is the Koszul complex on f_1^ n, \ldots , f_ r^ n over \mathcal{O}.
Proof.
In More on Algebra, Lemma 15.29.6 we have seen that the extended alternating Čech complex
\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}
is a colimit of the Koszul complexes K^ n = K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) sitting in degrees 0, \ldots , r. Note that K^ n is a finite chain complex of finite free \mathcal{O}-modules with dual \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K^ n, \mathcal{O}) = K_ n where K_ n is the Koszul cochain complex sitting in degrees -r, \ldots , 0 (as usual). By Lemma 52.6.8 the functor E \mapsto E^\wedge is gotten by taking R\mathop{\mathcal{H}\! \mathit{om}}\nolimits from the extended alternating Čech complex into E:
E^\wedge = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathop{\mathrm{colim}}\nolimits K^ n, E)
This is equal to R\mathop{\mathrm{lim}}\nolimits (E \otimes _\mathcal {O}^\mathbf {L} K_ n) by Cohomology on Sites, Lemma 21.48.8.
\square
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