  The nature of the relationship between low-level supermassive black hole
(SMBH) activity and galactic cold gas, if any, is currently unclear. Here, we
test whether central black holes may feed at higher rates in gas-rich galaxies,
probing SMBH activity well below the active regime down to Eddington ratios of
~1e-7. We use a combination of radio data from the ALFALFA survey and from the
literature, along with archival X-ray flux measurements from the Chandra X-ray
observatory, to investigate this potential relationship. We construct a sample
of 129 late-type galaxies, with MB<-18 out to 50 Mpc, that have both HI masses
and sensitive X-ray coverage. Of these, 75 host a nuclear X-ray source, a 58%
detection fraction. There is a highly significant correlation between nuclear
X-ray luminosity LX and galaxy stellar mass Mstar with a slope of 1.7+/-0.3,
and a tentative correlation (significant at the 2.8 sigma level) between LX and
HI gas mass MHI. However, a joint fit to LX as a function of both Mstar and MHI
finds no significant dependence on MHI (slope 0.1+/00.3), and similarly the
residuals of LX-LX(Mstar) show no trend with MHI; the apparent correlation
between LX and MHI seems to be entirely driven by Mstar. We demonstrate
quantitatively that these results are robust against X-ray binary
contamination. We conclude that the galaxy-wide cold gas content in these
spirals does not strongly correlate with their low-level supermassive black
hole activity, and suggest fueling is a localized process.
