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:
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:
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 .