Let $\alpha$ be a differential form of degree $p+1$ on $M\times\mathbb{R}$, where $M$ is an arbitrary smooth manifold, and $p$ a non negative integer.
Can $\alpha$ be always written as $\beta+\gamma\wedge dt$, where $\beta$ and $\gamma$ are differential forms on $M\times\mathbb{R}$ of degree $p+1$ and $p$ respectively, such that $\frac{\partial}{\partial t}\lrcorner\beta=0$ and $\frac{\partial}{\partial t}\lrcorner\gamma=0$? and, in the affirmative case, are $\beta$ and $\gamma$ uniquely determined?