[Abridged] We present the first results of hydrodynamical simulations that follow the formation of galaxies to z=0 in spherical regions of radius ~20 Mpc/h drawn from the Millennium Simulation. The regions have overdensities that deviate by (-2, -1, 0, +1, +2)sigma from the cosmic mean, where sigma is the rms mass fluctuation on a scale of ~20Mpc/h at z=1.5. The simulations have mass resolution of up to 10^6 Msun/h, cover the entire range of large-scale environments and allow extrapolation of statistics to the entire 500 (Mpc/h)^3 Millennium volume. They include gas cooling, photoheating from an ionising background, SNe feedback and winds, but no AGN. We find that the specific SFR density at z <~ 10 varies systematically from region to region by up to an order of magnitude, but the global value, averaged over all volumes, reproduces observational data. Massive, compact galaxies, similar to those observed in the GOODS fields, form in the overdense regions as early as z=6, but do not appear in the underdense regions until z~3. These environmental variations are not caused by a dependence of the star formation properties on environment, but rather by a strong variation of the halo mass function from one environment to another, with more massive haloes forming preferentially in the denser regions. At all epochs, stars form most efficiently in haloes of circular velocity ~ 250 km/s. However, the star formation history exhibits a form of downsizing (even in the absence of AGN): the stars comprising massive galaxies at z=0 have mostly formed by z=1-2, whilst those comprising smaller galaxies typically form at later times. However, additional feedback is required to limit star formation in massive galaxies at late times.
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