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Here goes a possibly simple question, but since I don't know the answer...

Consider a manifold $M$ and the differential graded algebra of forms on it:

\begin{equation} \Lambda(M) = \bigoplus_{k=0}^{n}\Lambda^{k}(M) \end{equation}

with $\Lambda^{0}(M)$ being the algebra of smooth functions on $M$. My question: is $\Lambda(M)$ generated by $\Lambda^{0}(M)$ and $\Lambda^{1}(M)$, in the sense that every form $\omega \in \Lambda(M)$ is equal to a sum of wedge products of elements of $\Lambda^{0}(M)$ and $\Lambda^{1}(M)$ ? If so, how can I prove it? If not, to which extent is something of that sort true?

Thanks!

[edited for typos]

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  • $\begingroup$ Have you tried proving it? Where do you get stuck? $\endgroup$ Commented Jan 29, 2018 at 18:07
  • $\begingroup$ Not in depth, but I imagine a proof could begin with an atlas of the manifold, and proving it locally... but then I wouldn't know how to extend that beyond a particular chart of the atlas... $\endgroup$ Commented Jan 29, 2018 at 19:02
  • $\begingroup$ Well, I suggest you try! (I am always baffled when people ask questions and when one asked what they try they say something like "well, I imagine one could..." For questions above a certain level —and certainly differential forms on a manifold are above that level— I personally find that extremely demotivating for providing an answer) $\endgroup$ Commented Jan 29, 2018 at 19:05
  • $\begingroup$ And I find your answer extremely discouraging, as a simple "yes, what you state is true" or "no, what you state isn't true" would suffice. If you find it discouraging to answer, simply don't. If you don't know the answer, then don't reply at all. $\endgroup$ Commented Jan 29, 2018 at 20:00
  • $\begingroup$ I know the answer, and I am pretty sure that if you know what the objects you mention in the question are you can probably answer it yourself. Trust me: as someone who's taught this subject quite a few times I know pretty well that the giving you the answer is the most damaging possible answer I could give you. Don't imagine that you could prove this locally: actually try to do it. If that works, then you can start thinking how to pass to the global situation. You can ask for hints with that, and I would probably give you one, if you explained what you had at that point. $\endgroup$ Commented Jan 29, 2018 at 20:03

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