The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero.
The irrelevant ideal J generated by all the Xi corresponds to the empty set, since not all homogeneous coordinates can vanish at a point of projective space.
Detailed studies, for example of canonical curves and the equations defining abelian varieties, show the geometric interest of systematic techniques to handle these cases.
The subject also grew out of elimination theory in its classical form, in which reduction modulo I is supposed to become an algorithmic process (now handled by Gröbner bases in practice).
Since this complex is intrinsic to R, one may define the graded Betti numbers βi, j as the number of grade-j images coming from Fi (more precisely, by thinking of φ as a matrix of homogeneous polynomials, the count of entries of that homogeneous degree incremented by the gradings acquired inductively from the right).
For this is considered as graded module over the homogeneous coordinate ring of the projective space, and a minimal free resolution taken.
[6] For curves Green showed that condition Np is satisfied when deg(L) ≥ 2g + 1 + p, which for p = 0 was a classical result of Guido Castelnuovo.