Differential forms vector space over function field?

Question

Let $V$ be a vector space and $V^*$ be its dual space. Then I know that $V^*$ is considered a vector space because we can scale the basis covectors by real numbers and add them together and all of the vector space axioms hold. But when we generalize the dual space to the space of differential forms, we don't just scale by numbers anymore. We scale by scalar functions. For instance we can define $\omega: = (x^2+1)dx$. The "scalar" in front of the basis diff form here is a function, not just a number. So do the differential forms actually form a vector space? If so, is it over some field of functions (maybe differentiable functions)? Also if anyone has a reference where this is discussed in detail that'd be great. Thanks.

Answer 1

Each cotangent space, i.e. the set of the 1-forms restricted to a point $x_0$ is a vector space. The set of the differential 1-forms over a manifold is more generally a module over the ring of the smooth functions: as you correctly pointed out, you can multiply differential forms by functions, not just by scalars. Nonetheless they form in particular also a vector space.