The coeval AGN and galaxy evolution and the observed local relations between SMBHs and galaxy properties suggest some connection or feedback between SMBH growth and galaxy build-up. We looked for correlations between properties of X-ray detected AGN and their FIR detected host galaxies, to find quantitative evidences for this connection, highly debated in the latest years. We exploit the rich multi-wavelength data set 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 (intrinsic luminosity and nuclear obscuration), and broad-band SED fitting to derive host galaxy properties (stellar mass $M_*$ and star formation rate SFR). We find that the AGN 2-10 keV luminosity ($L_{rm X}$) and the host $8-1000~mu m$ star formation luminosity ($L_{rm IR}^{rm SF}$) are significantly correlated. However, the average host $L_{rm IR}^{rm SF}$ has a flat distribution in bins of AGN $L_{rm X}$, while the average AGN $L_{rm X}$ increases in bins of host $L_{rm IR}^{rm SF}$, with logarithmic slope of $sim0.7$, in the redshifts range $0.4<z<1.2$. We also discuss the comparison between the distribution of these two quantities and the predictions from hydro-dynamical simulations. Finally we find that the average column density ($N_H$) shows a positive correlation with the host $M_*$, at all redshifts, but not with the SFR (or $L_{rm IR}^{rm SF}$). This translates into a negative correlation with specific SFR. Our results are in agreement with the idea that BH accretion and SF rates are correlated, but occur with different variability time scales. The presence of a positive correlation between $N_H$ and host $M_*$ suggests that the X-ray $N_H$ is not entirely due to the circum-nuclear obscuring torus, but may also include a contribution from the host galaxy.
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