The Stacks project

60.20 Divided power Poincaré lemma

Just the simplest possible version.

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. $\square$

An alternative (more direct) approach to the following lemma is explained in Example 60.25.2.

Lemma 60.20.2. Let $A$ be a ring. Let $(B, I, \delta )$ be a divided power ring. Let $P = B\langle x_ i \rangle $ be a divided power polynomial ring over $B$ with divided power ideal $J = IP + B\langle x_ i \rangle _{+}$ as usual. Let $M$ be a $B$-module endowed with an integrable connection $\nabla : M \to M \otimes _ B \Omega ^1_{B/A, \delta }$. Then the map of de Rham complexes

\[ M \otimes _ B \Omega ^*_{B/A, \delta } \longrightarrow M \otimes _ P \Omega ^*_{P/A, \delta } \]

is a quasi-isomorphism. Let $D$, resp. $D'$ be the $p$-adic completion of $B$, resp. $P$ and let $\Omega ^ i_ D$, resp. $\Omega ^ i_{D'}$ be the $p$-adic completion of $\Omega ^ i_{B/A, \delta }$, resp. $\Omega ^ i_{P/A, \delta }$. Let $M$ be a $p$-adically complete $D$-module endowed with an integral connection $\nabla : M \to M \otimes ^\wedge _ D \Omega ^1_ D$. Then the map of de Rham complexes

\[ M \otimes ^\wedge _ D \Omega ^*_ D \longrightarrow M \otimes ^\wedge _ D \Omega ^*_{D'} \]

is a quasi-isomorphism.

Proof. Consider the decreasing filtration $F^*$ on $\Omega ^*_{B/A, \delta }$ given by the subcomplexes $F^ i(\Omega ^*_{B/A, \delta }) = \sigma _{\geq i}\Omega ^*_{B/A, \delta }$. See Homology, Section 12.15. This induces a decreasing filtration $F^*$ on $\Omega ^*_{P/A, \delta }$ by setting

\[ F^ i(\Omega ^*_{P/A, \delta }) = F^ i(\Omega ^*_{B/A, \delta }) \wedge \Omega ^*_{P/A, \delta }. \]

We have a split short exact sequence

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

and the last module is free on $\text{d}x_ i$. It follows from this that $F^ i(\Omega ^*_{P/A, \delta }) \to \Omega ^*_{P/A, \delta }$ is a termwise split injection and that

\[ \text{gr}^ i_ F(\Omega ^*_{P/A, \delta }) = \Omega ^ i_{B/A, \delta } \otimes _ B \Omega ^*_{P/B, \delta } \]

as complexes. Thus we can define a filtration $F^*$ on $M \otimes _ B \Omega ^*_{B/A, \delta }$ by setting

\[ F^ i(M \otimes _ B \Omega ^*_{P/A, \delta }) = M \otimes _ B F^ i(\Omega ^*_{P/A, \delta }) \]

and we have

\[ \text{gr}^ i_ F(M \otimes _ B \Omega ^*_{P/A, \delta }) = M \otimes _ B \Omega ^ i_{B/A, \delta } \otimes _ B \Omega ^*_{P/B, \delta } \]

as complexes. By Lemma 60.20.1 each of these complexes is quasi-isomorphic to $M \otimes _ B \Omega ^ i_{B/A, \delta }$ placed in degree $0$. Hence we see that the first displayed map of the lemma is a morphism of filtered complexes which induces a quasi-isomorphism on graded pieces. This implies that it is a quasi-isomorphism, for example by the spectral sequence associated to a filtered complex, see Homology, Section 12.24.

The proof of the second quasi-isomorphism is exactly the same. $\square$

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