On submodules of vector fields I don't know much about modules aside from their basic definition and that they are more complicated than vector spaces.
I am asking this question because I wish to have a more "algebraic" understanding of Frobenius' theorem in differential geometry.
It seems to me that Frobenius' theorem can be stated using submodules of vector fields and ideals of the exterior algebra without necessarily invoking distributions.
If we do invoke distributions, then suppose that $M$ is a smooth $n$ dimensional manifold, $\Delta:x\mapsto\Delta_x$ is a $k$-dimensional distribution in $M$, and $\Gamma_M(\Delta)$ is the set of smooth sections of the distribution. Then $\Gamma_M(\Delta)\subseteq\mathfrak X(M)$ is a submodule of $\mathfrak X(M)$ the module of all (smooth vector fields) on $M$. Some statements can be made about this submodule:


*

*If $\Delta$ has a fixed dimension $k$, then $\Gamma_M(\Delta)$ is "locally finitely generated and of rank $k$", where I put quotes on this statement as I have no idea how this is called in the literature if at all, but what I mean is that around any $x\in M$ there is an open set $U$ such that the restriction $\Gamma_U(\Delta)$ is generated by $k$ pointwise linearly independent vector fields $E_1,...,E_k$, and this number $k$ is is minimal in the sense that there are no open sets (aside from $\varnothing$ I guess) where there are less linearly independent generators, like, say $F_1,...,F_l$ where $l<k$.

*I assume that if $\Delta$ has no fixed dimension, and its elements have $\dim\Delta_x\le k$, then every $x\in M$ has an open neighborhood $U$ where one can define $k$ generators, and not less in general, but these will not be pointwise linearly independent at all points.

*Let us call this submodule $\Gamma_M(\Delta)$ differentially closed if $X,Y\in\Gamma_M(\Delta)$ implies $[X,Y]\in\Gamma_M(\Delta)$. Frobenius' theorem is closely related to the fact - at least when $\Delta$ is of fixed dimension - that a differentially closed submodule (of locally rank $k$) always has $k$ commuting local generators.
The only proofs I know of Frobenius' theorem is valid only for the case of distributions with fixed dimension, where the existence of local, pointwise independent generators of a fixed number is used heavily in the proof.
Moreover, if we assume that $\Gamma_M(\Delta)$ is locally finitely generated and of rank $k$, then one can easily show that the set $\mathcal I\subseteq\Omega(M)$ consisting of differential forms that vanish when all of their arguments are elements of $\Gamma_M(\Delta)$ is an ideal of the exterior algebra that is locally generated by $n-k$ pointwise independent $1$-forms, the ideal is differential ($\mathrm d\mathcal I\subseteq I$) if and only if $\Gamma_M(\Delta)$ is differentially closed, and in this case $\mathcal I$ is always locally generated by $n-k$ exact $1$-forms.

Essentially, I am interested in statements (in-answer explanations, papers, textbooks) about submodules of $\mathfrak X(M)$ where these submodules are not defined by distributions from the get-go, such as


*

*Given an arbitrary submodule $\mathfrak M\le\mathfrak X(M)$, when does it have a well-defined "local rank"?

*Does a general submodule always determine a (possibly degenerate) distribution?

*How does duality with an ideal of the exterior algebra work in the general case? I relied heavily on local generators for proving dual statements. In particular, if there is always a corresponding distribution then I guess the differential closedness of the submodule/ideal implies complete integrability of the distribution, but what about the statements about commuting/exact generators?

*Etc.
I don't expect a complete answer here, but at least some paper or advanced textbook that treats Frobenius' theorem from such a heavily algebraic point of view would be nice and illuminating for me.
 A: H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. A.M.S., vol. 180, June 1973, discusses these issues at length. Any module of vector fields has a notion of orbit, a minimal nonempty subset closed under the flows of the vector fields, and the orbits are submanifolds, but perhaps of different dimensions. If the module is closed under its own flows, and of constant rank, the Frobenius theorem applies. Consider the module of all smooth vector fields on the plane $f(x,y)\partial_x+g(x,y)\partial_y$ so that $g=0$ for $x<0$. These vector fields have as orbits the entire plane, so a local picture of the module involved is never enough. Clearly there is no local rank defined near the vertical axis. For relations between differential form descriptions of the Frobenius theorem and the vector fields descriptions, you can see p. 6 of my lecture notes on exterior differential systems.
A: For the second question, since a distribution is a subbundle of the tangent bundle, and therefore a vector bundle itself, if the manifold is compact and Hausdorff, you need the submodule $\mathfrak{M}$ to be a projective module and finitely generated. This is because of Serre-Swan theorem. I think that this paper could be very useful for you (see section 3).
Sorry for this highly incomplete answer.
