  Observations of neutral hydrogen (HI) and molecular gas show that 50% of all
nearby early-type galaxies (ETGs) contain some cold gas. Molecular gas is
always found in small gas discs in the central region of the galaxy, while
neutral hydrogen is often distributed in a low-column density disc or ring
typically extending well beyond the stellar body. Dust is frequently found in
ETGs as well. The goal of our study is to understand the link between dust and
cold gas in nearby ETGs as a function of HI content. We analyse deep optical
$g-r$ images obtained with the MegaCam camera at the Canada-France-Hawaii
Telescope for a sample of 21 HI-rich and 41 HI-poor ETGs. We find that all
HI-rich galaxies contain dust seen as absorption. Moreover, in 57 percent of
these HI-rich galaxies, the dust is distributed in a large-scale spiral
pattern. Although the dust detection rate is relatively high in the HI-poor
galaxies ($\sim$59 percent), most of these systems exhibit simpler dust
morphologies without any evidence of spiral structures. We find that the
HI-rich galaxies possess more complex dust morphology extending to almost two
times larger radii than HI-poor objects. We measured the dust content of the
galaxies from the optical colour excess and find that HI-rich galaxies contain
six times more dust (in mass) than HI-poor ones. In order to maintain the dust
structures in the galaxies, continuous gas accretion is needed, and the
substantial HI gas reservoirs in the outer regions of ETGs can satisfy this
need for a long time. We find that there is a good correspondence between the
observed masses of the gas and dust, and it is also clear that dust is present
in regions further than 3~Reff. Our findings indicate an essential relation
between the presence of cold gas and dust in ETGs and offer a way to study the
interstellar medium in more detail than what is possible with HI observations.
