next up previous
Next: exterior algebras Up: connection Previous: direct sums, tensor products,

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

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

Then we may define a connection

$\displaystyle \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

$\displaystyle \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$

ARRAY(0x9522fb8)


2012-02-29