Context : The coeval active galactic nuclei ( AGN ) and galaxy evolution and the observed local relations between super massive black holes ( SMBHs ) and galaxy properties suggest some sort of connection or feedback between the SMBH growth ( i.e. , the AGN activity ) and the galaxy build-up ( i.e. , the star formation history ) .

Aims : We have looked for correlations between average properties of X-ray detected AGN and their FIR detected , star forming host galaxies , in order to find quantitative evidences for this connection , that has been highly debated in the latest years .

Methods : We exploit the rich multi-wavelength data set ( from X-ray to FIR ) available in the COSMOS field for a large sample ( 692 sources ) of AGN and their hosts , in the redshift range 0.1 < z < 4 .
We use X-ray data to select AGN and determine their properties , such as X-ray intrinsic luminosity and nuclear obscuration , and broad-band ( from UV to FIR ) SED fitting results to derive host galaxy properties , such as stellar mass ( M _ { * } ) and star formation rate ( SFR ) .

Results : We find that the AGN 2-10 keV luminosity ( L _ { X } ) and the host 8 - 1000 ~ { } \mu m star formation luminosity ( L _ { IR } ^ { SF } ) are significantly correlated , even after removing the dependency of both quantities with redshift .
However , the average host L _ { IR } ^ { SF } has a flat distribution in bins of AGN L _ { X } , while the average AGN L _ { X } increases in bins of host L _ { IR } ^ { SF } with logarithmic slope of \sim 0.7 , in the redshifts range 0.4 < z < 1.2 .
We also discuss the comparison between the full distribution of these two quantities and the predictions from hydrodynamical simulations .
No other significant correlations between AGN L _ { X } and host properties is found .
On the other hand , we find that the average column density ( N _ { H } ) shows a clear positive correlation with the host M _ { * } , at all redshifts , but not with the SFR ( or L _ { IR } ^ { SF } ) .
This translates into a negative correlation with specific SFR , at all redshifts .
The same is true if the obscured fraction is computed .

Conclusions : Our results are in agreement with the idea , introduced in recent galaxy evolutionary models , that BH accretion and SF rates are correlated , but occur with different variability time scales .
Finally , the presence of a positive correlation between N _ { H } and host M _ { * } suggests that the column density that we observe in the X-rays is not entirely due to the circum-nuclear obscuring torus , but may also include a significant contribution from the host galaxy .