Dust temperature in ALMA $\hbox{[C $\scriptstyle\rm II $]}$-detected
  high-$z$ galaxies
  At redshift $z>5$ the far-infrared (FIR) continuum spectra of main-sequence
galaxies are sparsely sampled, often with a single data point. The dust
temperature $T_{\rm d, SED}$ thus has to be assumed in the FIR continuum
fitting. This introduces large uncertainties regarding the derived dust mass
($M_{\rm d}$), FIR luminosity, and obscured fraction of the star formation
rate. These are crucial quantities to quantify the effect of dust obscuration
in high-$z$ galaxies. To overcome observations limitations, we introduce a new
method that combines dust continuum information with the overlying $\hbox{[C
$\scriptstyle\rm II $]} 158\mu$m line emission. By breaking the $M_{\rm d} -
T_{\rm d, SED}$ degeneracy, with our method, we can reliably constrain the dust
temperature with a single observation at $158\mu$m. This method can be applied
to all ALMA and NOEMA $\hbox{[C $\scriptstyle\rm II $]}$ observations and
exploited in ALMA Large Programs such as ALPINE and REBELS targeting $\hbox{[C
$\scriptstyle\rm II $]}$ emitters at high-$z$. We also provide a physical
interpretation of the empirical relation recently found between $molecular$ gas
mass and $\hbox{[C $\scriptstyle\rm II $]}$ luminosity. We derive an analogous
relation linking the $total$ gas surface density and $\hbox{[C $\scriptstyle\rm
II $]}$ surface brightness. By combining the two, we predict the cosmic
evolution of the surface density ratio $\Sigma_{\rm H_2} / \Sigma_{\rm gas}$.
We find that $\Sigma_{\rm H_2} / \Sigma_{\rm gas}$ slowly increases with
redshift, which is compatible with current observations at $0 < z < 4$.
