A new definition of the sheath edge is rigorously derived taking into account the kinetic properties of the plasma, and a consistent set of local sheath edge conditions is presented for the case of a magnetic field perpendicular to the wall. These local boundary conditions give explicit expressions for the ion velocity, the electron velocity, and the electron heat flux at the sheath edge, which can be easily implemented in a fluid code. It is shown that in the case of positive current to the wall, the commonly used Bohm's relations well aproximate the proposed boundary conditions, while large discrepancies are observed for negative currents. A fully kinetic PIC code simulating the plasma wall transition has been developed to validate these local relations, showing an excellent agreement with the theory. This work represents a first step towards a complete formulation of the sheath edge local boundary conditions for a general magnetic geometry.