  We present the results of our Keck/DEIMOS spectroscopic follow-up of
candidate galaxies of i-band-dropout protocluster candidate galaxies at
$z\sim6$ in the COSMOS field. We securely detect Lyman-$\alpha$ emission lines
in 14 of the 30 objects targeted, 10 of them being at $z=6$ with a
signal-to-noise ratio of $5-20$, the remaining galaxies are either
non-detections or interlopers with redshift too different from $z=6$ to be part
of the protocluster. The 10 galaxies at $z\approx6$ make the protocluster one
of the riches at $z>5$. The emission lines exhibit asymmetric profiles with
high skewness values ranging from 2.87 to 31.75, with a median of 7.37. This
asymmetry is consistent with them being Ly$\alpha$, resulting in a redshift
range of $z=5.85-6.08$. Using the spectroscopic redshifts, we re-calculate the
overdensity map for the COSMOS field and find the galaxies to be in a
significant overdensity at the $4\sigma$ level, with a peak overdensity of
$\delta=11.8$ (compared to the previous value of $\delta=9.2$). The
protocluster galaxies have stellar masses derived from Bagpipes SED fits of
$10^{8.29}-10^{10.28} \rm \,M_{\rm \odot}$ and star formation rates of
$2-39\,\rm M_{\rm \odot}\rm\,yr^{-1}$, placing them on the main sequence at
this epoch. Using a stellar-to-halo-mass relationship, we estimate the dark
matter halo mass of the most massive halo in the protocluster to be $\sim
10^{12}\rm M_{\rm \odot}$. By comparison with halo mass evolution tracks from
simulations, the protocluster is expected to evolve into a Virgo- or Coma-like
cluster in the present day.
