  We establish generalizations of Saito's criterion for the freeness of
divisors in projective spaces that apply both to sequences of several
homogeneous polynomials and to divisors on other complete varieties. As an
application, the new criterion is applied to several examples, including
sequences whose polynomials depend on disjoint sets of variables, some
sequences that are equivariant for the action of a linear group, blow-ups of
divisors, and certain sequences of polynomials in positive characteristics.
