On the lifting problem for homogeneous ideals

The lifting problem for homogeneous ideals is studied. A relation between a homogeneous ideal J and its liftings is established using a syzygy basis of J. This relation is then used to obtain an algorithm for finding all the liftings of a homogeneous ideal. As an application of the algorithm, we discover the first example of a homogeneous ideal of dimension 0 in four variables which is not liftable to a radical ideal over the field of rational numbers.