Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)We begin the study of a tilting theory in certain truncated categories of modules G(Gamma) for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where Gamma = P+ x J, J is an interval in Z, and P+ is the set of dominant integral weights of the simple Lie algebra. We use this to put a tilting theory on the category G(Gamma') where Gamma' = P' x J, where P' subset of P+ is saturated. Under certain natural conditions on Gamma', we note that G(Gamma') admits full tilting modules.10Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP [2012/06923-0, 2011/22322-4]