For a smooth, $n$-dimensional manifold $M$ over $\mathbb{R}$, I would like to know if sections of the dual space of the order-$k$ jet bundle sit within a commutative differential graded algebra structure. I'm imagining that this would restrict to the De-Rham complex when $n = 1$.
More specifically, for order $n$, what I am naïvely imagining is something like $C^\infty(M) \otimes \Lambda( dx^I)$, where $I$ ranges over all multi-indices $I = (I_1, I_2, \ldots, I_n) \in \mathbb{N}^n $ where $\sum I_j < k$ and $dx^I$ is the 'dual' of the section $$ \dfrac{\partial^{\sum I_j}}{dx_1^{I_1}\ldots dx_n^{I_n}}$$ of the order-$k$ jet bundle. My guess is the exterior derivative in degree $0$ would just sum up over all the $<k$-th order partials, then be extended via the Leibniz rule.
Does anyone know
- if there is an obvious problem with this idea, or if there exists something like this that has been well studied?
- if it computes singular cohomology of the manifold, like the De-Rham case.
Thanks!