Remark 60.6.9 (07KJ)—The Stacks project

Remark 60.6.9. Consider a commutative diagram of rings

\[ \xymatrix{ B \ar[r]_\varphi & B' \\ A \ar[u] \ar[r] & A' \ar[u] } \]

Let $\Omega _{B/A} \to \Omega $ and $\Omega _{B'/A'} \to \Omega '$ be quotients satisfying the assumptions of Algebra, Lemma 10.132.1. Assume there is a map $\varphi : \Omega \to \Omega '$ which fits into a commutative diagram

\[ \xymatrix{ \Omega _{B/A} \ar[r] \ar[d] & \Omega _{B'/A'} \ar[d] \\ \Omega \ar[r]^{\varphi } & \Omega ' } \]

where the top horizontal arrow is the canonical map $\Omega _{B/A} \to \Omega _{B'/A'}$ induced by $\varphi : B \to B'$. In this situation, given any pair $(M, \nabla )$ where $M$ is a $B$-module and $\nabla : M \to M \otimes _ B \Omega $ is a connection we obtain a base change $(M \otimes _ B B', \nabla ')$ where

\[ \nabla ' : M \otimes _ B B' \longrightarrow (M \otimes _ B B') \otimes _{B'} \Omega ' = M \otimes _ B \Omega ' \]

is defined by the rule

\[ \nabla '(m \otimes b') = \sum m_ i \otimes b'\text{d}\varphi (b_ i) + m \otimes \text{d}b' \]

if $\nabla (m) = \sum m_ i \otimes \text{d}b_ i$. If $\nabla $ is integrable, then so is $\nabla '$, and in this case there is a canonical map of de Rham complexes (Remark 60.6.8)

60.6.9.1

\begin{equation} \label{crystalline-equation-base-change-map-complexes} M \otimes _ B \Omega ^\bullet \longrightarrow (M \otimes _ B B') \otimes _{B'} (\Omega ')^\bullet = M \otimes _ B (\Omega ')^\bullet \end{equation}

which maps $m \otimes \eta $ to $m \otimes \varphi (\eta )$.

Comments (2)

Comment #4914 by Rubén Muñoz--Bertrand on February 06, 2020 at 03:16

At the beginning aren't we supposed to read $\Omega_{B'/A'}\to\Omega'$ instead of $\Omega_{B'/A}\to\Omega'$ ?

Comment #5184 by Johan on June 10, 2020 at 11:47

Thanks and fixed here.

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