The incidence and star-formation properties of X-ray active galactic nuclei across the star-forming main sequence in DESI-eRASS1: the role of host-galaxy morphology

The interplay between star formation and supermassive black-hole growth is central to galaxy evolution, but how host-galaxy morphology regulates star-formation enhancement and AGN triggering across the star-forming main sequence remains unclear. We investigate the star-formation properties and incidence of X-ray AGN across the star-forming main sequence using the DESI-eRASS1 dataset, focusing on host structure. Our analysis includes 1171 X-ray selected AGN and 45374 non-AGN star-forming galaxies at $z\le1.5$. We quantify star formation in AGN hosts relative to matched control samples using ${\rm SFR}_{norm}$ and examine its dependence on X-ray luminosity ($L_X$) and specific accretion rate ($\lambda_{sBHAR}$). We measure AGN incidence as a function of distance from the star-forming main sequence ($\Delta_{\rm MS}$), separating disk- and spheroid-dominated systems based on Sersic index. ${\rm SFR}_{norm}$ remains close to unity at low to intermediate $L_X$ and increases at higher luminosities, with the transition shifting toward higher $L_X$ in more massive systems. This trend depends on morphology: in the stellar-mass range $10.5\le\log(M_\star/M_\odot)<11.5$, disk-dominated AGN hosts exhibit enhanced ${\rm SFR}_{norm}$ at moderate $L_X$, while spheroid-dominated systems remain consistent with unity. No comparable morphology dependence is found when using $\lambda_{sBHAR}$. The incidence of X-ray AGN increases strongly with $\Delta_{\rm MS}$ in both redshift bins, with a steeper dependence at higher redshift. The $\Delta_{\rm MS}$-incidence relation is also morphology-dependent and evolves with redshift. The connection between star formation and AGN activity is governed by global gas availability but modulated by host-galaxy structure and cosmic epoch. Absolute AGN power is more tightly linked to host-wide star formation than accretion efficiency normalised by stellar mass.