The Stacks project

Proposition 15.92.2. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. The functor which sends $K \in D(A)$ to the derived limit $K' = R\mathop{\mathrm{lim}}\nolimits ( K \otimes _ A^\mathbf {L} A/I^ n )$ is the left adjoint to the inclusion functor $D_{comp}(A) \to D(A)$ constructed in Lemma 15.90.10.

Proof. Say $(f_1, \ldots , f_ r) = I$ and let $K_ n^\bullet $ be the Koszul complex with respect to $f_1^ n, \ldots , f_ r^ n$. By Lemma 15.90.18 it suffices to prove that

\[ R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} K_ n^\bullet ) = R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} A/(f_1^ n, \ldots , f_ r^ n) ) = R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} A/I^ n ). \]

By Lemma 15.92.1 the pro-objects $\{ K_ n^\bullet \} $ and $\{ A/(f_1^ n, \ldots , f_ r^ n)\} $ of $D(A)$ are isomorphic. It is clear that the pro-objects $\{ A/(f_1^ n, \ldots , f_ r^ n)\} $ and $\{ A/I^ n\} $ are isomorphic. Thus the map from left to right is an isomorphism by Lemma 15.86.12. $\square$


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