  We investigate the cool gas absorption in galaxy clusters by
cross-correlating MgII absorbers detected in quasar spectra from Data Release
16 of the Sloan Digital Sky Survey (SDSS) with galaxy clusters identified in
the Dark Energy Spectroscopic Instrument (DESI) survey. We find significant
covering fractions ($1-5\, \%$ within $r_{500}$, depending on the chosen
redshift interval), $\sim 4-5$ times higher than around random sightlines.
While the covering fraction of cool gas in clusters decreases with increasing
mass of the central galaxy, the total MgII mass within $r_{\rm 500}$ is
nonetheless $\sim 10$ times higher than for SDSS luminous red galaxies (LRGs).
The MgII covering fraction versus impact parameter is well described by a power
law in the inner regions and a exponential function at larger distances. The
characteristic scale of the transition between these two regimes is smaller for
large equivalent width absorbers. Cross-correlating MgII absorption with
photo-z selected cluster member galaxies from DESI reveals a statistically
significant connection. The median projected distance between MgII absorbers
and the nearest cluster member is $\sim200$ kpc, compared to $\sim500$ kpc in
random mocks with the same galaxy density profiles. We do not find a
correlation between MgII strength and the star formation rate of the closest
cluster neighbour. This suggests that cool gas in clusters, as traced by MgII
absorption, is: (i) associated with satellite galaxies, (ii) dominated by cold
gas clouds in the intracluster medium, rather than by the interstellar medium
of galaxies, and (iii) may originate in part from gas stripped from these
cluster satellites in the past.
