We make use of the dual number
. That means, we
consider an algebra
.
We define
We assume
(An equivalent and (probably) easier way to describe
This method is also valid when we deal with the interior derivation.)
Let us then define a map
by the following formula.
We may easily see that the map
by
is a
-module homomorphism.
So
together defines an algebra homomorphism
For any 1-form
So
We decompose the homomorphism above as
and obtain the exterior derivation
for any non negative integer
The theory of exterior derivation may of course be generalized to
a theory of that on a separated scheme
over a scheme
.