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Next: curvature of connections Up: exterior derivation Previous: exterior derivation on 1-forms

exterior derivation on general forms

We make use of the dual number $ \epsilon$ . That means, we consider an algebra $ A_\epsilon=A[\epsilon]/(\epsilon^2)$ . We define

$\displaystyle E= A_\epsilon\otimes_A (\wedge \Omega^1_{B/A}).
$

We assume $ \epsilon$ is odd. That means, we equip an $ A$ -algebra structure on $ E$ by introducing the following commutation relations.

$\displaystyle \epsilon \omega =-\omega \epsilon \quad (\forall \omega \in \Omega^1_{B/A}.)
$

(An equivalent and (probably) easier way to describe $ E$ is given by considering a free $ B$ -module $ B\epsilon$ . We then define

$\displaystyle E=\wedge_B (B\epsilon \oplus \Omega^1_{B/A}).
$

This method is also valid when we deal with the interior derivation.)

Let us then define a map

$\displaystyle \phi_0:B \to E
$

by the following formula.

$\displaystyle \phi_0(x)=(1+\epsilon d)(x)=x+\epsilon d x.
$

We may easily see that the map $ \phi_0$ is an algebra homomorphism. We regard $ E$ as a $ B$ -algebra via this homomorphism. We then define

$\displaystyle \phi_1:\Omega^1_{B/A}\to E
$

by

$\displaystyle \phi_1(\omega)=(1+\epsilon d)\omega.
$

$ \phi_1$ is a $ B$ -module homomorphism.

So $ \phi_0,\phi_1$ together defines an algebra homomorphism

$\displaystyle \phi:\otimes_A \Omega^1_{B/A} \to E.
$

For any 1-form $ \omega \in \Omega^1_{B/A}$ , we have

$\displaystyle \phi(\omega \otimes \omega )
=(\omega + \epsilon d \omega )\wedge(\omega +\epsilon d \omega )
=0.
$

So $ \phi$ factors through the exterior algebra and define an algebra homomorphism

$\displaystyle \phi_{(\bullet)}:\wedge_A \Omega^1_{B/A} \to E.
$

We decompose the homomorphism above as $ \phi_{(\bullet)}=1+\epsilon d$ and obtain the exterior derivation

$\displaystyle d: \Omega^k_{B/A} \to \Omega^{k+1}_{B/A}
$

for any non negative integer $ k$ . It is easy to verify that the exterior derivation $ d$ satisfies the rules (EXT1) and (EXT2).

The theory of exterior derivation may of course be generalized to a theory of that on a separated scheme $ X$ over a scheme $ S$ .


next up previous
Next: curvature of connections Up: exterior derivation Previous: exterior derivation on 1-forms
2012-02-29