  We extend the adiabatic regularization method to spin-1/2 fields. The ansatz
for the adiabatic expansion for fermionic modes differs significantly from the
WKB-type template that works for scalar modes. We give explicit expressions for
the first adiabatic orders and analyze particle creation in de Sitter
spacetime. As for scalar fields, the adiabatic method can be distinguished by
its capability to overcome the UV divergences of the particle number operator.
We also test the consistency of the extended method by working out the
conformal and axial anomalies for a Dirac field in a FLRW spacetime, in exact
agreement with those obtained from other renormalization prescriptions. We
finally show its power by computing the renormalized stress-energy tensor for
Dirac fermions in de Sitter space.
