The Stacks project

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 21.46.8. $\square$


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