The Algebra of Differential Forms
We’ve defined the exterior bundle 
over a manifold 
. Given any open 
we’ve also defined a 
-form over 
to be a section of this bundle: a function 
such that 
. We write 
for the collection of all such -forms over . It’s straightforward to see that this defines a sheaf on .
This isn’t just a sheaf of sets; it’s a sheaf of modules over the structure sheaf 
of smooth functions on . We define the necessary operations pointwise:
&=\alpha(p)+\beta(p)\\\left[f\alpha\right](p)&=f(p)\alpha(p)\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5B%5Calpha%2B%5Cbeta%5Cright%5D%28p%29%26%3D%5Calpha%28p%29%2B%5Cbeta%28p%29%5C%5C%5Cleft%5Bf%5Calpha%5Cright%5D%28p%29%26%3Df%28p%29%5Calpha%28p%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left[\alpha+\beta\right](p)&=\alpha(p)+\beta(p)\\\left[f\alpha\right](p)&=f(p)\alpha(p)\end{aligned}")
where the right hand sides are defined by the vector space structures on the respective 
.
We can go even further and define the sheaf of differential forms
This sheaf 
is not just a sheaf of modules over , it’s a sheaf of algebras. For an 
and a 
, we define their exterior product pointwise:
=\alpha(p)\wedge\beta(p)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cleft%5B%5Calpha%5Cwedge%5Cbeta%5Cright%5D%28p%29%3D%5Calpha%28p%29%5Cwedge%5Cbeta%28p%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left[\alpha\wedge\beta\right](p)=\alpha(p)\wedge\beta(p)")
In fact, this is a graded algebra, and the multiplication has degree zero:
Even better, this is a unital algebra. We see this by considering the zero grade, since the unit must live in the zero grade. Indeed, 
, so sections of 
are simply functions on . That is, 
. Given a function 
we will just write 
instead of 
.
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July 12, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
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