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The Stacks project

Lemma 15.91.17. In Situation 15.91.15. Let K \in D(A). The following are equivalent

  1. K is derived complete with respect to I, and

  2. the canonical map K \to R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} K_ n^\bullet ) is an isomorphism of D(A).

Proof. If (2) holds, then K is derived complete with respect to I by Lemma 15.91.16. Conversely, assume that K is derived complete with respect to I. Consider the filtrations

K_ n^\bullet \supset \sigma _{\geq -r + 1}K_ n^\bullet \supset \sigma _{\geq -r + 2}K_ n^\bullet \supset \ldots \supset \sigma _{\geq -1}K_ n^\bullet \supset \sigma _{\geq 0}K_ n^\bullet = A

by stupid truncations (Homology, Section 12.15). Because the construction R\mathop{\mathrm{lim}}\nolimits (K \otimes E) is exact in the second variable (Lemma 15.87.11) we see that it suffices to show

R\mathop{\mathrm{lim}}\nolimits \left( K \otimes _ A^\mathbf {L} (\sigma _{\geq p}K_ n^\bullet / \sigma _{\geq p + 1}K_ n^\bullet ) \right) = 0

for p < 0. The explicit description of the Koszul complexes above shows that

R\mathop{\mathrm{lim}}\nolimits \left( K \otimes _ A^\mathbf {L} (\sigma _{\geq p}K_ n^\bullet / \sigma _{\geq p + 1}K_ n^\bullet ) \right) = \bigoplus \nolimits _{i_1, \ldots , i_{-p}} T(K, f_{i_1}\ldots f_{i_{-p}})

which is zero for p < 0 by assumption on K. \square


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