The Stacks project

Lemma 60.19.3. With notation as in ( and (, given any cosimplicial module $M_*$ over $D(*)$ and $i > 0$ the cosimplicial module

\[ M_0 \otimes ^\wedge _{D(0)} \Omega ^ i_{D(0)} \to M_1 \otimes ^\wedge _{D(1)} \Omega ^ i_{D(1)} \to M_2 \otimes ^\wedge _{D(2)} \Omega ^ i_{D(2)} \to \ldots \]

is homotopic to zero, where $\Omega ^ i_{D(n)}$ is the $p$-adic completion of the $i$th exterior power of $\Omega _{D(n)}$.

Proof. By Lemma 60.19.2 the endomorphisms $0$ and $1$ of $\Omega _{D(*)}$ are homotopic. If we apply the functor $\wedge ^ i$ we see that the same is true for the cosimplicial module $\wedge ^ i\Omega _{D(*)}$, see Lemma 60.16.1. Another application of the same lemma shows the $p$-adic completion $\Omega ^ i_{D(*)}$ is homotopy equivalent to zero. Tensoring with $M_*$ we see that $M_* \otimes _{D(*)} \Omega ^ i_{D(*)}$ is homotopic to zero, see Lemma 60.16.1 again. A final application of the $p$-adic completion functor finishes the proof. $\square$

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