  We study phenomenology of a light scalar dark matter (DM). In the model,
there are an inert doublet scalar and a singlet Dirac fermion $\psi$, both
charged under a global $Z_2$ symmetry. The mass of the lightest inert scalar
$H$ can be lighter than 10 GeV by imposing appropriate relations between three
scalar quartic couplings. The lightest $Z_2$ odd particle is stable and DM. In
this paper, focusing on the parameter space where $H$ is lighter than $\psi$
and is DM, we discuss DM physics related to relic density, direct detection,
indirect detection, collider searches and other cosmological observations. We
clarify differences from the case where $\psi$ is instead DM, which has been
focused on in the previous works.
