The Stacks project

Lemma 15.94.7. Let $I$ be an ideal in a Noetherian ring $A$. Let ${}^\wedge $ denote derived completion with respect to $I$. Let $K \in D^-(A)$.

  1. If $M$ is a finite $A$-module, then $(K \otimes _ A^\mathbf {L} M)^\wedge = K^\wedge \otimes _ A^\mathbf {L} M$.

  2. If $L \in D(A)$ is pseudo-coherent, then $(K \otimes _ A^\mathbf {L} L)^\wedge = K^\wedge \otimes _ A^\mathbf {L} L$.

Proof. Let $L$ be as in (2). We may represent $K$ by a bounded above complex $P^\bullet $ of free $A$-modules. We may represent $L$ by a bounded above complex $F^\bullet $ of finite free $A$-modules. Since $\text{Tot}(P^\bullet \otimes _ A F^\bullet )$ represents $K \otimes _ A^\mathbf {L} L$ we see that $(K \otimes _ A^\mathbf {L} L)^\wedge $ is represented by

\[ \text{Tot}((P^\bullet )^\wedge \otimes _ A F^\bullet ) \]

where $(P^\bullet )^\wedge $ is the complex whose terms are the usual $=$ derived completions $(P^ n)^\wedge $, see for example Proposition 15.94.2 and Lemma 15.94.6. This proves (2). Part (1) is a special case of (2). $\square$

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