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Given a compact, smooth manifold $M$ and a real vector bundle $E \to M$ (in general not flat). There already have been numerous questions about how to equip the space $\bigoplus_k \Gamma(\Lambda^k T^* M \otimes E) = : \Gamma(\Lambda^\bullet T^* M \otimes E)$ with a nontrivial generalization of the de Rham differential making the space into a complex. This, in general, only seems nicely possible if $E$ admits a flat connection, see for example here and here.

Now, I have a situation in which I know what I want my cohomology to be, but I do not know a good candidate for what my differential could be. To be precise, if $V$ is the standard fiber of $E$, I would like to equip the space $\Gamma(\Lambda^\bullet T^* M \otimes E)$ with a differential $d : \Gamma(\Lambda^\bullet T^* M \otimes E) \to \Gamma(\Lambda^{\bullet+1} T^* M \otimes E)$, which firstly should be a differential operator of order 1 and secondly should give the cohomology $$ H^k(\Gamma(\Lambda^{\bullet} T^* M \otimes E)) \cong H_{\text{dR}}^k(M) \otimes_{\mathbb{R}} V.$$

I don't necessarily need this differential to have anything to do with the de Rham differential on $M$ directly, and you're free to use connections. Is there a more or less well-known candidate fulfilling this property? Or is this doomed to fail if we do not allow for flat connections? Also if the question is underspecified in some way, let me know.

The reason I believe a differential like this should exist is because, by my understanding, it should show up in the last step of Theorem 2.4.1a in Fuks' book "Cohomology of infinite-dimensional Lie algebras". The details are omitted, but I think a differential as I describe needs to exist in some way for their result to hold. If needed I can give some more details.

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    $\begingroup$ You should put some conditions on these differentials, or it must be completely useless to you. Because all of the individual terms $C^k$ are infinite dimensional, you can cook up a differential with essentially arbitrary homology, so long as it is concentrated in degrees $0$ through $\dim M$. $\endgroup$
    – mme
    Commented Sep 3, 2019 at 14:06
  • $\begingroup$ Yeah, that is a good point. I think a reasonable requirement in my context should be that the differential is a differential operator of order 1. I will add this to the post. Thank you! $\endgroup$
    – user126256
    Commented Sep 3, 2019 at 14:11
  • $\begingroup$ I don't immediately see where Fuks' book uses what you are looking for. Can you point to the precise step you are thinking of? $\endgroup$
    – mme
    Commented Sep 6, 2019 at 11:44
  • $\begingroup$ I think I understand my issue now; on page 147 of the book, Fuks says that the first page differential $d_1^{p,q}$ is induced by the de Rham differential. However, if you track the calculations back a bit, this differential would have to come from a de Rham differential on the space $\Gamma(\Lambda^\bullet T^* M \otimes E)$ for some other vector bundle $E$. He does not mention (but other papers by Gelfand, Fuks and Losik regarding this matter do), that this bundle $Q$ is flat, and even trivializable. Thus my question becomes redundant for my purposes. Thank you regardless! $\endgroup$
    – user126256
    Commented Sep 9, 2019 at 14:13

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