We present an analysis of star formation and nuclear activity in galaxies as a function of both luminosity and environment in the fourth data release of the Sloan Digital Sky Survey. Using a sample of 27753 galaxies in the redshift range 0.005 < z < 0.037 that is >~90 per cent complete to Mr = -18.0, we find that the Hα equivalent width, EW(Hα), distribution is strongly bimodal, allowing galaxies to be robustly separated into passively evolving and star-forming populations about a value EW(Hα) = 2 Å. In high-density regions ~70 per cent of galaxies are passively evolving independent of luminosity. In the rarefied field, however, the fraction of passively evolving galaxies is a strong function of luminosity, dropping from 50 per cent for Mr <~ -21 to zero by Mr ~ -18. Indeed for the lowest luminosity range covered (-18 < Mr < -16) none of the ~600 galaxies in the lowest-density quartile is passively evolving. The few passively evolving dwarf galaxies in field regions appear as satellites to bright (>~ L*) galaxies. We find a systematic reduction of ~30 per cent in the Hα emission from dwarf (-19 < Mr < -18) star-forming galaxies in high-density regions with respect to field values, implying that the bulk of star-forming dwarf galaxies in groups and clusters are currently in the process of being slowly transformed into passive galaxies. The fraction of galaxies with the optical signatures of an active galactic nucleus (AGN) decreases steadily from ~50 per cent at Mr ~ -21 to ~0 per cent by Mr ~ -18 closely mirroring the luminosity dependence of the passive galaxy fraction in low-density environments. This result reflects the increasing importance of AGN feedback with galaxy mass for their evolution, such that the star formation histories of massive galaxies are primarily determined by their past merger history. In contrast, the complete absence of passively evolving dwarf galaxies more than ~2 virial radii from the nearest massive halo (i.e. cluster, group or massive galaxy) indicates that internal processes, such as merging, AGN feedback or gas consumption through star formation, are not responsible for terminating star formation in dwarf galaxies. Instead the evolution of dwarf galaxies is primarily driven by the mass of their host halo, probably through the combined effects of tidal forces and ram-pressure stripping.
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