The method introduced by Vojta to give a different proof of Mordell's conjecture has been generalized by Faltings to establish Lang's conjecture on abelian varieties and then further extended by Vojta to deal with semi-abelian varieties. In each case, the heart of the proof can be summarized in an inequality of heights, obtained via diophantine approximation. Here, we present a generalization of this step. We show it is not necessary to work with algebraic groups for this part and phrase our theorem only in terms of sheaves on projective schemes. This allows us to introduce more parameters in the statement and offers a wider range of applications. In the semi-abelian case, we obtain a variation of Vojta's result which implies Poonen's conjecture. Even in the abelian case, our inequality leads to strengthenings of Faltings' theorem.