Wedge with a closed form preserves exactness classes

Given: Closed forms $\alpha ,\beta$ of degrees $p,q\ge 0$, and forms $\eta ,\theta$ of the indicated degrees.

Closed and exact differential forms: For the complex def-de-rham-cochain-complex, put $Z^k(M)=\ker (d:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k+1}(M))$ and $B^k(M)=\mathrm{im}(d:{\mathrm{\Omega }}^{k-1}(M)\to {\mathrm{\Omega }}^k(M))$. A form is closed if it belongs to $Z^k$ and exact if it belongs to $B^k$. If $\omega =d\eta$, then $d\omega =d^2\eta =0$ by thm-the-exterior-derivative-squares-to-zero, so $B^k\subseteq Z^k$. In particular $B^0=0$, since ${\mathrm{\Omega }}^{-1}=0$. The zero form is both closed and exact in every degree.

The exterior derivative is a graded derivation: Let $M$ be a smooth manifold. The exterior derivative is an $\mathbb{R}$-linear map $d:{\mathrm{\Omega }}^{\ast }(M)\to {\mathrm{\Omega }}^{\ast }(M)$ of degree one. For homogeneous smooth forms $\alpha \in {\mathrm{\Omega }}^p(M)$ and $\beta \in {\mathrm{\Omega }}^q(M)$, $d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1{)}^{\mathrm{deg}\alpha }\alpha \wedge d\beta \mathrm{.}$