We investigate the chemical properties of local galaxies within a cosmological framework in the hierarchical picture of galaxy formation. To this aim, we use a hierarchical semi-analytic model which includes the contribution from (i) low and intermediate mass stars, (ii) type Ia Supernovae (SNe) and (iii) massive stars. - Abridged - We compare our predictions with available observations in the Milky Way (MW), in local dwarf galaxies and in local ellipticals. For Milky-Way-like galaxies, we can successfully reproduce the [O-Fe] vs [Fe/H] relation observed in disc stars and the stellar metallicity distribution (SMD). For dwarf galaxies, the stellar metallicity vs mass relation is reproduced by assuming that a substantial fraction of the heavy elements is lost through metal-enhanced outflows and a type Ia SN realization probability lower than the one of MW-like galaxies. - Abridged - In ellipticals, the observations indicate higher [alpha/Fe] values in larger galaxies. - Abridged - Our results computed with a standard Salpeter initial mass function (IMF) indicate a flat [alpha/Fe]-sigma relation. However, we suggest a possible solution to this problem and show how, by assuming a star formation-dependent IMF with a slope x=1.35 in systems with star formation rates