We present a simple method for deriving the renormalization counterterms from
the components of the energy-momentum tensor in curved space-time. This method
allows full control over the finite parts of the counterterms and provides
explicit expressions for each term separately. As an example, the method is
used for the self-interacting scalar field in a Friedmann-Robertson-Walker
metric in the adiabatic approximation, where we calculate the renormalized
equation of motion for the field and the renormalized components of the
energy-momentum tensor to fourth adiabatic order while including interactions
to one-loop order. Within this formalism the trace anomaly, including
contributions from interactions, is shown to have a simple derivation. We
compare our results to those obtained by two standard methods, finding
agreement with the Schwinger-DeWitt expansion but disagreement with adiabatic
subtractions for interacting theories.Comment: 25 pages, published versio
