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 ith 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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