Suppose that $\partial$ is a non-trivial ($\partial \neq 0$) cohomology operator on an $m-$dimensional manifold $M$ (that is: $\partial:\Omega(M)\to\Omega(M)$ is a degree $1$ derivation such that $\partial^2=0$). What additional conditions are needed to guarantee that there exists a local basis of $1-$forms of the form $\partial f_1,\ldots ,\partial f_m$, with $f_i\in C^\infty (M)$ for $1\leq i\leq m$.?
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$\begingroup$ What do you mean by local basis here? If you just want a basis of $\Omega^1(M)$ of the given type, then it seems like $H^1(\Omega(M),\partial)=0$ is a necessary and sufficient condition. $\endgroup$– Mark GrantCommented Apr 7, 2015 at 17:11
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$\begingroup$ Sorry for the slopiness in the question. What I mean is that the functions $f_i$ could be different for each open subset $U\subset M$, and for each $p\in U$, $\{\partial f_1(p),\ldots,\partial f_m(p)$ must be a basis of $T^*_p M$. $\endgroup$– GrimolattoCommented Apr 7, 2015 at 18:52
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$\begingroup$ Also, notice that, for the case $\partial =\mathrm{d}$, $H^1(S^1,\mathrm{d})=\mathbb{R}$... $\endgroup$– GrimolattoCommented Apr 7, 2015 at 19:27
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