Confusion about cohomologous differential forms

Question

Two forms are cohomologous if they differ by an exact form, so for $\alpha, \alpha'$ in the same cohomology class we may say $\alpha - \alpha' = d\beta$. Do we know in general the degree of $d\beta$ or of $\beta$? Say $\alpha$ and $\alpha'$ are $p$ forms, can we say $d\beta$ is a $p-1$-form or something of the sort?

Answer 1

If $\alpha$ and $\alpha'$ are $p$-forms, then $d\beta=\alpha-\alpha'$ is a $p$-form (sum of two $p$-forms is a $p$-form). Since $d\beta$ is a $p$-form, then $\beta$ must be a $(p-1)$-form, for the exterior differentiation $d$ maps $k$ forms to $(k+1)$-forms.