The Stacks project

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. $\square$

Comments (4)

Comment #4023 by Nils Waßmuth on

I think there is a typo, the split exact sequence should give information on the graded pieces of and not on the graded pieces of

Comment #6267 by Mingchen on

In the statement of the lemma, should be instead.

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