to the active AGN phase, we require the model to match not only the observables concerning the galaxy population, but also those concerning the AGN population, in order to have a reliable modelization for the interplay between the cooling and star formation processes and the growth of SMBH with the ensuing AGN emission and feedback.

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[Fig. 1. - The relation between the black hole mass $M_{BH}$ and the one-dimensional velocity dispersion $\sigma$ of the bulges of the host model galaxies at $z=0$ is compared with data from Ferrarese & Merritt (2000, filled squares) and Gebhardt et al. (2000, circles); the color code refers to logarithm of the abundance normalized to its maximum value. ]{}

Thus, we first show in fig. 1 the $m_{BH}-\sigma$ relation, to check that, when integrated over time, the BH accretion that we implement in our model is consistent with the local observations for the whole range of galactic mass spanned by the model. Note that the spread in the $m_{BH}-\sigma$ increases for decreasing galactic mass, a feature shared by all hierarchical models and related to the increasing number of merging histories involved the formation of massive haloes, which results in a lower statistical deviation. Note 