  The first generation of stars (commonly known as population III) are expected
to form in low-mass protogalaxies in which molecular hydrogen is the dominant
coolant. Radiation from these stars will rapidly build up an extragalactic
ultraviolet background capable of photodissociating H2, and it is widely
believed that this background will suppress further star formation in low-mass
systems.
  However, star formation will also produce an extragalactic X-ray background.
This X-ray background, by increasing the fractional ionization of protogalactic
gas, promotes H2 formation and reduces the effectiveness of ultraviolet
feedback.
  In this paper, we examine which of these backgrounds has the dominant effect.
Using a simple model for the growth of the UV and X-ray backgrounds, together
with a detailed one-dimensional model of protogalactic chemical evolution, we
examine the effects of the X-ray backgrounds produced by a number of likely
source models. We show that in several cases, the resulting X-ray background is
strong enough to offset UV photodissociation in large H2-cooled protogalaxies.
On the other hand, small protogalaxies (those with virial temperatures T_vir <
2000K) remain dominated by the UV background in all of the models we examine.
  We also briefly investigate the effects of the X-ray background upon the
thermal and chemical evolution of the diffuse IGM.
