The Global Dimension of $\omega$-Smash Coproducts

We mainly study the global dimension of $\omega$-smash coproducts. We show that if $H$ is a Hopf algebra with a bijective antipode $S_H$, and $C{_\omega}\bowtie H$ denotes the $\omega$-smash coproduct, then
$$\mathrm{gl}.\mathrm{dim}(C_\omega\bowtie H)\leq \mathrm{gl}.\mathrm{dim}(C)+\mathrm{gl}.\mathrm{dim}(H),$$
where $\mathrm{gl}.\mathrm{dim}(H)$ denotes the global dimension of $H$ as a coalgebra.