Remark 59.6.8. Let $A \to B$ be a ring map. Let $\Omega _{B/A} \to \Omega $ be a quotient satisfying the assumptions of Algebra, Lemma 10.132.1. Let $M$ be a $B$-module. A *connection* is an additive map

such that $\nabla (bm) = b \nabla (m) + m \otimes \text{d}b$ for $b \in B$ and $m \in M$. In this situation we can define maps

by the rule $\nabla (m \otimes \omega ) = \nabla (m) \wedge \omega + m \otimes \text{d}\omega $. This works because if $b \in B$, then

As is customary we say the connection is *integrable* if and only if the composition

is zero. In this case we obtain a complex

which is called the de Rham complex of the connection.

## Comments (2)

Comment #4223 by Dario Weißmann on

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