Macaulay's lemma states:
Let R be a polynomial ring and I a homogeneous ideal. Then the Hilbert function of I is the same as the Hilbert function of in(I).
(Schenck, Computational Algebraic Geometry, p55)
(Where in(I) / lt(I) is the ideal consisting of leading terms of elements of I, and by the Hilbert function of I we mean the Hilbert function of R/I.)
Is there a counterexample showing that this isn't necessarily true for inhomogeneous ideals? (In all the cases that I have tried, it seems also to be true for inhomogeneous ideals.)