The Stacks project

Lemma 60.20.1. Let $A$ be a ring. Let $P = A\langle x_ i \rangle $ be a divided power polynomial ring over $A$. For any $A$-module $M$ the complex

\[ 0 \to M \to M \otimes _ A P \to M \otimes _ A \Omega ^1_{P/A, \delta } \to M \otimes _ A \Omega ^2_{P/A, \delta } \to \ldots \]

is exact. Let $D$ be the $p$-adic completion of $P$. Let $\Omega ^ i_ D$ be the $p$-adic completion of the $i$th exterior power of $\Omega _{D/A, \delta }$. For any $p$-adically complete $A$-module $M$ the complex

\[ 0 \to M \to M \otimes ^\wedge _ A D \to M \otimes ^\wedge _ A \Omega ^1_ D \to M \otimes ^\wedge _ A \Omega ^2_ D \to \ldots \]

is exact.

Proof. It suffices to show that the complex

\[ E : (0 \to A \to P \to \Omega ^1_{P/A, \delta } \to \Omega ^2_{P/A, \delta } \to \ldots ) \]

is homotopy equivalent to zero as a complex of $A$-modules. For every multi-index $K = (k_ i)$ we can consider the subcomplex $E(K)$ which in degree $j$ consists of

\[ \bigoplus \nolimits _{I = \{ i_1, \ldots , i_ j\} \subset \text{Supp}(K)} A \prod \nolimits _{i \not\in I} x_ i^{[k_ i]} \prod \nolimits _{i \in I} x_ i^{[k_ i - 1]} \text{d}x_{i_1} \wedge \ldots \wedge \text{d}x_{i_ j} \]

Since $E = \bigoplus E(K)$ we see that it suffices to prove each of the complexes $E(K)$ is homotopic to zero. If $K = 0$, then $E(K) : (A \to A)$ is homotopic to zero. If $K$ has nonempty (finite) support $S$, then the complex $E(K)$ is isomorphic to the complex

\[ 0 \to A \to \bigoplus \nolimits _{s \in S} A \to \wedge ^2(\bigoplus \nolimits _{s \in S} A) \to \ldots \to \wedge ^{\# S}(\bigoplus \nolimits _{s \in S} A) \to 0 \]

which is homotopic to zero, for example by More on Algebra, Lemma 15.28.5. $\square$

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