Local units versus local projectivity. Dualisations: Corings with local   structure maps
  We unify and generalize different notions of local units and local projectivity. We investigate the connection between these properties by constructing elementary algebras from locally projective modules. Dual versions of these constructions are discussed, leading to corings with local comultiplications, corings with local counits and rings with local multiplications. 
Joost Vercruysse