Lemma 61.20.1 (099M)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.20: Derived completion in the constant Noetherian case
Lemma 61.20.1 (cite)

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Lemma 61.20.1. Let $\mathcal{C}$ be a site. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. The left adjoint to the inclusion functor $D_{comp}(\mathcal{C}, \Lambda ) \to D(\mathcal{C}, \Lambda )$ of Algebraic and Formal Geometry, Proposition 52.6.12 sends $K$ to

\[ K^\wedge = R\mathop{\mathrm{lim}}\nolimits (K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}) \]

In particular, $K$ is derived complete if and only if $K = R\mathop{\mathrm{lim}}\nolimits (K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n})$.

Proof. Choose generators $f_1, \ldots , f_ r$ of $I$. By Algebraic and Formal Geometry, Lemma 52.6.9 we have

\[ K^\wedge = R\mathop{\mathrm{lim}}\nolimits (K \otimes _\Lambda ^\mathbf {L} \underline{K_ n}) \]

where $K_ n = K(\Lambda , f_1^ n, \ldots , f_ r^ n)$. In More on Algebra, Lemma 15.94.1 we have seen that the pro-systems $\{ K_ n\} $ and $\{ \Lambda /I^ n\} $ of $D(\Lambda )$ are isomorphic. Thus the lemma follows. $\square$

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