  The jets of radio AGN provide one of the most important forms of AGN
feedback, yet considerable uncertainties remain about how they are triggered.
Since the molecular gas reservoirs of the host galaxies can supply key
information about the dominant triggering mechanism(s), here we present Atacama
Large Millimeter/sub-millimeter Array (ALMA) CO(1-0) observations of a complete
sample of 29 powerful radio AGN ($P_{1.4GHz} > 10^{25}$ W Hz$^{-1}$ and $0.05 <
z < 0.3$) with an angular resolution of about 2 - 3 arcsec (corresponding to 2
- 8 kpc). We detect molecular gas with masses in the range $10^{8.9} < M_{H_2}
< 10^{10.2}$ M$_{\odot}$ in the early-type host galaxies of 10 targets, while
for the other 19 sources we derive upper limits. The detection rate of objects
with such large molecular masses -- $34\pm9$% -- is higher than in the general
population of non-active early-type galaxies (ETG: $<$10%). The kinematics of
the molecular gas are dominated in most cases by rotating disk-like structures,
with diameters up to 25 kpc. Compared with the results for samples of quiescent
ETG in the literature, we find a larger fraction of more massive, more extended
and less settled molecular gas structures. In most of the CO-detected sources,
the results are consistent with triggering of the AGN as the gas settles
following a merger or close encounter with a gas-rich companion. However, in a
minority of objects at the centres of rich clusters of galaxies, the accretion
of gas cooling from the hot X-ray halos is a plausible alternative to galaxy
interactions as a triggering mechanisms.
