Let $M$ be a n-dimensional closed smooth manifold, $\eta\_{k}$ be the k-form on $M$, $X$ be the smooth vector field on $M$.Let $\eta=\eta\_{0}+\eta\_{1}+\cdots+\eta\_{n-1}+\eta\_{n}$ be a ($d-i\_{X}$)-closed form,
then we have $$i\_{X}\eta\_{n}=d\eta\_{n-2},$$ the relation imply that $\eta\_{n}$ is exact outside the set $M\_{0}$ of zeros of the vector field $X$. How to get the result?


Edit:
I read this result in the chapter 7 of "Heat Kernels and Dirac Operators", page 203.
My question is why the relation imply that $\eta\_{n}$ is exact outside the set $M\_{0}$.



A question about equivariant closed form
If you can find a closed 1-form $\alpha\in \Omega^1(M\setminus Z(X))$ with $i\_X\alpha=1$, then $\eta\_n=\alpha\wedge i\_X\eta\_n = \alpha\wedge d\eta\_{n-2}= -d(\alpha\wedge\eta\_{n-2})$ is exact.
Maybe, the source that you are reading has such $\alpha$.


