Hilbert functions, residual intersections, and residually S-2 ideals

Let R be a homogeneous ring over an infinite field, I subset ofR a homogene ous ideal, and a subset ofI an ideal generated by s forms of degrees d(1),. ..,d(s) so that codim(a:I)greater than or equal tos. We give broad conditio ns for when the Hilbert function of R/a or of R/(a:I) is determined by I an d the degrees d(1),...,d(s). These conditions are expressed in terms of res idual intersections of I, culminating in the notion of residually S-2 ideal s. We prove that the residually S-2 property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce id eals with this property.