  It is well established that the [alpha/Fe] ratios in elliptical galaxies
increase with galaxy mass. This relation holds also for early-type dwarf
galaxies, although it seems to steepen at low masses. The [alpha/Fe] vs. mass
relation can be explained assuming that smaller galaxies form over longer
timescales (downsizing), allowing a larger amount of Fe (mostly produced by
long-living Type Ia Supernovae) to be released and incorporated into newly
forming stars. Another way to obtain the same result is by using a flatter
initial mass function (IMF) in large galaxies, increasing in this way the
number of Type II Supernovae and therefore the production rate of
alpha-elements. The integrated galactic initial mass function (IGIMF) theory
predicts that the higher the star formation rate, the flatter the IMF. We have
checked, by means of semi-analytical calculations, that the IGIMF theory,
combined with the downsizing effect (i.e. the shorter duration of the star
formation in larger galaxies), well reproduces the observed [alpha/Fe] vs. mass
relation. In particular, we show a steepening of this relation in dwarf
galaxies, in accordance with the available observations.
