A search for CII-158$\mu$m line emission in HCM\,6A, a Lyman-$\alpha$
  emitter at $z=6.56$
  We report a Plateau de Bure interferometer search for CII-158$\mu$m emission
from HCM6A, a lensed Lyman-$\alpha$ emitter (LAE) at $z = 6.56$. Our
non-detections of CII-158$\mu$m line emission and 1.2mm radio continuum
emission yield $3\sigma$ limits of L$_{\rm CII} < 6.4 \times 10^7 \times
(\Delta V/100 km s^{-1})^{1/2}$ L$_\odot$ on the CII-158$\mu$m line luminosity
and S$_{\rm 1.2mm} < 0.68$ mJy on the 1.2mm flux density. The local conversion
factor between L$_{\rm CII}$ and star formation rate (SFR) yields an SFR $<
4.7$ M$_\odot$ yr$^{-1}$, $\approx 2$ times lower than that inferred from the
ultraviolet (UV) continuum, suggesting that the local factor may not be
applicable in high-$z$ LAEs. The non-detection of 1.2mm continuum emission
yields a total SFR $< 28$ M$_\odot$/yr; any obscured star formation is thus
within a factor of two of the visible star formation. Our best-fit model to the
rest-frame UV/optical spectral energy distribution of HCM6A yields a stellar
mass of $1.3 \times 10^9$ M$_\odot$ and an SFR of ~10 M$_\odot$/yr, with
negligible dust obscuration. We fortuitously detect CO J=3-2 emission from a
$z=0.375$ galaxy in the foreground cluster Abell370, obtaining a CO line
luminosity of L$^\prime ({\rm CO}) > (8.95 \pm 0.79) \times 10^8$ K km s$^{-1}$
pc$^2$, and a molecular gas mass of M$({\rm H_2}) > (4.12 \pm 0.36) \times
10^9$ M$_\odot$, for a CO-to-H$_2$ conversion factor of 4.6 M$_\odot$ (K km
s$^{-1}$ pc$^2$)$^{-1}$.
