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connections extended

Let $ X$ be a separated scheme over a scheme $ S$ . Let $ \mathcal{V}$ be a quasi coherent sheaf over $ X$ with a connection

$\displaystyle \nabla : \mathcal{V}\to \Omega^1_{X/S} \otimes_{\mathcal{O}_X} \mathcal{V}
$

on it. We extend $ \nabla$ to

$\displaystyle \nabla:\Omega^\bullet_{X/S} \otimes_{\mathcal{O}_X}\mathcal{V}
\to \Omega^\bullet_{X/S} \otimes_{\mathcal{O}_X} \mathcal{V}.
$

by defining

$\displaystyle \nabla(\alpha\otimes v)
=(d\alpha )\otimes v + (-1)^k \alpha \wed...
...abla v)
\qquad (\forall \alpha \in \Omega^k_{X/S}, \forall v \in \mathcal{V}).
$

In other words,

$\displaystyle \nabla \vert _{\Omega^k_{X/S}\otimes \mathcal{V}}=
d\otimes \operatorname{id}+(-1)^k \operatorname{id}\wedge \nabla.
$

It is easy to verify that $ \nabla$ is well-defined and satisfy

$\displaystyle \nabla(\alpha\wedge x)= (d \alpha) \wedge x
+ (-1)^k \alpha \wed...
...{X/S}, \forall x \in
\Omega^\bullet_{X/S}\otimes_{\mathcal{O}_X}\mathcal{V}).
$



2012-02-29