Nonlocal lattice fermion models on the 2d torus

Abelian fermion models described by the SLAC action are considered on a finite 2d lattice. It is shown that modification of these models by introducing additional Pauli - Villars regularization supresses nonlocal effects and provides agreement with the continuum results in vectorial U(1) models. In the case of chiral fermions the phase of the determinant differs from the continuum one.