  We use a simple model of spheroid formation to explore the relationship
between the creation of stars and dust in a massive proto galaxy and the growth
of its central black hole. This model predicts that sub-mm luminosity peaks
after only ~ 0.2 Gyr. However, without a very massive seed black hole,
Eddington-limited growth means that a black hole mass of 10^9 solar masses, and
hence very luminous AGN activity, cannot be produced until > 0.5 Gyr after the
formation of the first massive stars in the halo. The model thus predicts a
time-lag between the peak of sub-mm luminosity and AGN luminosity in a massive
proto-elliptical of a few times 10^8 years. For a formation redshift z = 5,
this means that powerful AGN activity is delayed until z = 3.5, by which time
star formation in the host is 90% complete, and sub-mm luminosity has declined
to ~ 25% of its peak value. This provides a natural explanation for why
successful sub-mm detections of luminous radio galaxies are largely confined to
z > 2.5. Conversely the model also predicts that while all high-redshift
luminous sub-mm-selected sources should contain an active (and growing) black
hole, the typical luminosity of the AGN in such objects is ~1000 times smaller
than that of the most powerful AGN. This is consistent with the almost complete
failure to detect sub-mm selected galaxies with existing X-ray surveys. Finally
the model yields a black-hole:spheroid mass ratio which evolves rapidly in the
first Gyr, but asymptotes to ~ 0.001-0.003 in agreement with results at low
redshift. This ratio arises not because the AGN terminates star formation, but
because fueling of the massive black hole is linked to the total mass of gas
available for star formation in the host.
