  The decoupling of heavy fields as required by the Appelquist-Carazzone
theorem plays a fundamental role in the construction of any effective field
theory. However, it is not a trivial task to implement a renormalization
prescription that produces the expected decoupling of massive fields, and it is
even more difficult in curved spacetime. Focused on this idea, we consider the
renormalization of the one-loop effective action for the Yukawa interaction
with a background scalar field in curved space. We compute the beta functions
within a generalized DeWitt-Schwinger subtraction procedure and discuss the
decoupling in the running of the coupling constants. For the case of a
quantized scalar field, all the beta function exhibit decoupling, including
also the gravitational ones. For a quantized Dirac field, decoupling appears
almost for all the beta functions. We obtain the anomalous result that the mass
of the background scalar field does not decouple.
