  We calculate the correlation energy of a two-dimensional homogeneous electron
gas using several available approximations for the exchange-correlation kernel
$f_{\rm xc}(q,\omega)$ entering the linear dielectric response of the system.
As in the previous work of Lein {\it et al.} [Phys. Rev. B {\bf 67}, 13431
(2000)] on the three-dimensional electron gas, we give attention to the
relative roles of the wave number and frequency dependence of the kernel and
analyze the correlation energy in terms of contributions from the $(q,
i\omega)$ plane. We find that consistency of the kernel with the electron-pair
distribution function is important and in this case the nonlocality of the
kernel in time is of minor importance, as far as the correlation energy is
concerned. We also show that, and explain why, the popular Adiabatic Local
Density Approximation performs much better in the two-dimensional case than in
the three-dimensional one.
