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. ∗ firstname.lastname@example.org Supported by an INFN fellowship. e-mail: email@example.com Among the increasing variety of 2D dilaton gravity models, a particular attention is certainly deserved by spherically symmetric reduced General Relativity (GR) in interaction to minimally coupled massless 4D scalar fields (first considered in ). Because of its direct link to the reald 4D world, this model has been regarded as a reliable device to investigate four dimensional physics in the s-wave sector and not just as a “laboratory” like other 2D models. The relevant effective action (or, better, a part of it) for the matter sector can be directly obtained by functional integration of the 2D conformal anomaly . This action is nonlocal, as it involves the inverse of the Delambertian operator. It can be made local by the introduction of two auxiliary fields . Applying a technique which has already been tested on an analogous problem in 4D , in this paper we will investigate vacuum polarization around a Schwarzschild black hole induced by quantum minimal scalar fields using the above mentioned effective action. Starting from the local form of it we will impose appropriate boundary conditions to the auxiliary fields in order to select the relevant quantum states, namely Boulware (vacuum polarization around a static star), Unruh (black hole evaporation) and Israel-Hartle-Hawking (thermal equilibrium). The resulting expectation values of the stress tensor will be then compared to the ones obtained by canonical quantization and Hadamard regularization (see for instance ). As we shall see and as expected on the basis of previous results (,  and references therein), this check dramatically fails and once again puts serious doubts on the possibility of inferring the actual 4D behaviour starting from this lower dimensional effective action. While a similar procedure has been discussed in  for the Israel-Hartle-Hawking case, here the analysis is more general and extended to Boulware and, in the most nontrivial case, to the Unruh state.