The use of lower-dimensional models is exploited in many areas of physics as a way to simplify the mathematical treatment of very complicated phenomena while, at the same time, retaining the main physical ingredients. For the case of black holes, the Hawking evaporation process1 can be understood using a simple model where the gravity action is coupled to quantized matter in the form of free two-dimensional minimal scalar fields. In particular, the effective action one derives by straightforward integration of the trace anomaly2 gives a stress energy tensor which in the Schwarzschild spacetime perfectly agrees with the results one gets by standard canonical quantization. Trying to improve the model, i.e. by employing a spherically reduced 4d minimal scalar field3, one faces a number of difficulties. Unphysical results obtained for the evaporation of black holes (such as antievaporation3,4) using the anomaly induced effective action led us to perform a rigorous calculation of the quantum stress tensor in Schwarzschild by using the point-splitting regularization technique5: exact asymptotic results close to the horizon and at infinity have been derived in the three quantum states of interest (namely Boulware, Hartle-Hawking and Unruh) and two analytic approximations proposed , one valid for large r (based on the WKB) and the other being physically meaningful in the region close to the horizon. Finally, based on these results a "phenomenological" modification of the (unsatisfactory) anomaly induced effective action has been carried out6.