  We study the co-evolution between the black hole accretion rate (BHAR) and
the star formation rate (SFR) in different galaxy life phases: main sequence
star-forming galaxies, quiescent and starburst galaxies at different cosmic
epochs.
  We take advantage of the X-ray data from the Chandra COSMOS-Legacy survey and
of the extensive multiwavelength ancillary observations in the COSMOS field
presented in the COSMOS2015 catalog. We perform an X-ray stacking analysis and
combine it with detected sources, in a broad redshift interval ($0.1<z<3.5$).
The X-ray luminosity is used to predict the BHAR, while a similar stacking
analysis on far-infrared Herschel maps is used to measure the corresponding
SFR.
  We focus on the evolution of the average SFR-stellar mass (M*) relation and
compare it with the BHAR-M* relation. We find that the ratio between BHAR and
SFR does not evolve with redshift, although it depends on stellar mass. For the
star-forming populations, this dependence on M* has a logarithmic slope of
$\sim0.6$, for the starburst sample of $\sim0.4$, both at odds with quiescent
sources where it remains constant ($\log(\rm {BHAR}/{\rm SFR})\sim -3.4$). By
studying the specific BHAR and specific SFR we find signs of downsizing for
both M* and black hole mass (M$_{\rm BH}$): quiescents grew their super-massive
black hole at very early times, while star-forming and starburst galaxies had
an accretion that endured until more recent times.
  Our results support the idea that the same physical processes feed and
sustain both star formation and black hole accretion. Our integrated estimates
of the M*-M$_{\rm BH}$ relation at all redshifts are consistent with
independent determinations of the local M*-M$_{\rm BH}$ relation, thus adding
key evidence to a weak evolution in the BHAR/SFR, and its low normalization
compared to local dynamical M*-M$_{\rm BH}$ relations.
