  The destabilising effect of Hall diffusion in a Keplerian disc allows the MRI
to occur for much lower ionisation levels than would otherwise be possible.
However, simulations suggest that the consequences for the saturated state are
not as significant as suggested by the linear instability. Close inspection
reveals that that the simulations have not yet probed the Hall-dominated
regime. Here we revisit the effect of Hall diffusion on the MRI and the
implications for the extent of MHD turbulence in protoplanetary discs.
  We conduct a local stability analysis for a vertical, weak magnetic field
subject to axisymmetric perturbations with a vertical wave vector. The
diffusivity dependence is presented using analytic expressions and contours in
the eta_H - eta_P plane for the maximum growth rate and corresponding wave
number, the upper cut-off for unstable wave numbers, and the loci that divide
the plane into regions of different characteristic behaviour. In the
highly-diffusive limit the magnetic field decouples from the fluid
perturbations and the diffusive MRI reduces to a diffusive plane-parallel shear
instability with effective shear rate 1.5 Omega. We give analytic expressions
for the growth rate and wave number of the most unstable mode.
  Finally, we illustrate the critical effect of Hall diffusion on the extent of
dead zones in protoplanetary discs by applying a local stability criterion to a
simple model of the minimum-mass solar nebula at 1 au, including x-ray and
cosmic-ray ionisation and a population of 1 micron grains. Hall diffusion
increases or decreases the MRI-active column density by an order of magnitude
or more, depending on whether B is parallel or antiparallel to the rotation
axis, respectively. Existing estimates of the depth of magnetically active
layers in protoplanetary discs are likely to be wildly inaccurate.
  [Abridged]
