Conservation of the Dirac Current in Models with a General Spin Connection

Here I obtain the conditions necessary for the conservation of the Dirac current when one substitutes the assumption $γ^A_{\ \ |B}=0$ for $γ^A_{\ \ |B}=[V_B,γ^A]$, where the $γ^A$s are the Dirac matrices and "$|$" represents the components of the covariant derivative. As an application, I apply these conditions to the model used in Ref. [M. Novello, Phys. Rev. {\bf D8}, 2398 (1973)].