  Minimally coupled 4D scalar fields in Schwarzschild space-time are
considered. Dimensional reduction to 2D leads to a well known anomaly induced
effective action, which we consider here in a local form with the introduction
of auxiliary fields. Boundary conditions are imposed on them in order to select
the appropriate quantum states (Boulware, Unruh annd Israel-Hartle-Hawking).
The stress tensor is then calculated and its comparison with the expected 4D
form turns out to be unsuccessful. We also critically discuss in some detail a
recent controversial result appeared in the literature on the same topic.
