functoriality

LEMMA 1.2   Let $C$ be a commutative ring. Let $A,B$ be commutative $C$ -algebras. Let $\varphi:A\to B$ be a $C$ -algebra homomorphism. Assume we are given an $A$ -module $M$ with a connection

$$\nabla: M\to \Omega^1_{A/C} \otimes_A M.$$

Then we may define a connection

$$\nabla_{(B)} : B\otimes_A M \to \Omega^1_{B/C}\otimes_B (B\otimes_A M) =\Omega^1_{B/C}\otimes_A M$$

on a pullback $B\otimes_A M$ by defining

$$\nabla_{(B)}(b\otimes m)= d b \otimes m + b \cdot (d\varphi \otimes 1) \nabla(m)$$

for all $b\in B$ , $m\in M$ .

In fact, it is easy to see that the map $\nabla_{(B)}$ is well defined and that it satisfies the rule (Co) of connection. $\qedsymbol$

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