AGN feedback and the origin of the $\alpha$ enhancement in early type
  galaxies - insights from the GAEA model
  We take advantage of our recently published model for Galaxy Evolution and
Assembly (GAEA) to study the origin of the observed correlation between
[$\alpha$/Fe] and galaxy stellar mass. In particular, we analyse the role of
radio mode AGN feedback, that recent work has identified as a crucial
ingredient to reproduce observations. In GAEA, this process introduces the
observed trend of star formation histories extending over shorter time-scales
for more massive galaxies, but does not provide a sufficient condition to
reproduce the observed $\alpha$ enhancements of massive galaxies. In the
framework of our model, this is possible only assuming that any residual star
formation is truncated for galaxies more massive than $10^{10.5}\,{\rm
M}_{\odot}$. This results, however, in even shorter star formation time-scales
for the most massive galaxies, that translate in total stellar metallicities
significantly lower than observed. Our results demonstrate that (i) trends of
[$\alpha$/Fe] ratios cannot be simply converted in relative time-scale
indicators, and (ii) AGN feedback cannot explain alone the positive correlation
between [$\alpha$/Fe] and galaxy mass/velocity dispersion. Reproducing
simultaneously the mass-metallicity relation and the $\alpha$ enhancements
observed poses a challenge for hierarchical models, unless more exotic
solutions are adopted such as metal-rich winds or a variable IMF.
