Let $U \subseteq \mathbb{R}^n$ be an open subset.
Then there are two ways to define a differential form:
- algebraically as in this previous post, were one considers the real algebra $A$ of smooth functions $A$, the module of real derivations and then its exterior algebra;
- geometrically, as smooth maps $$\omega \colon U \to \operatorname{Alt}^p(\mathbb{R}^n).$$
This second approach is used in most differential geometry books,
such as Henri Cartan's Differential forms.
One possible advantage in the second approach is that one can
consider $C^k$ maps
$$\omega \colon U \subseteq E \to \operatorname{Alt}^p(E, F)$$
where $E$ and $F$ are Banach spaces. The apparent disadvantage is that
one has to do a lot of work to give an algebraic structure to those maps.
I have never seen a discussion on the relation between those two definitions so here are some natural questions:
- Is it trivial to go from algebraic to geometric definition and back
(in the particular case of $E = \mathbb{R}^n$, $F = \mathbb{R}$)? - Is there an equivalence / adjunction?
- Has this something to do with the relation between bundles and lf. modules of sheaves (the algebro-geometric duality)?
If i understand correctly, the relation should be
$$
\Lambda^p(\mathcal{D}^1(C^{\infty}(U, F))) =
C^{\infty}(U, \operatorname{Alt}^p(E, F))
$$
where one shows that the second gadget has a well defined wedge product.
The first gadget represents the alternating functor, hence
$$
\operatorname{Alt}^p(\mathcal{D}^1(C^{\infty}(U, F)), N) =
\operatorname{Hom}(C^{\infty}(U, \operatorname{Alt}^p(E, F)), N)
$$
for any $A$-module $N$ where $A = C^{\infty}(U, F)$.