  The stratification-driven Hall instability in a weakly ionized polytropic
plasma is investigated in the local approximation within an equilibrium
Keplerian disk of a small aspect ratio. The leading order of the asymptotic
expansions in the aspect ratio is applied to both equilibrium as well as the
perturbation problems. The equilibrium disk with an embedded purely toroidal
magnetic field is found to be stable to radial, and unstable to vertical
short-wave perturbations. The marginal stability surface is found in the space
of the local Hall and inverse plasma beta parameters, as well as the free
parameter of the model which is related to the total current through the disk.
To estimate the minimal values of the equilibrium magnetic field that leads to
instability, the latter is constructed as a sum of a current free magnetic
field and the simplest approximation for magnetic field created by a
distributed electric current.
