GENERIC BOUNDS FOR FROBENIUS CLOSURE AND TIGHT CLOSURE

Abstract

We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P = k[x(0), ... , x(d)], one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard-graded k-algebra R of dimension d 1. This indicates that the tight closure of an ideal behaves more uniformly than the ideal itself. Moreover, if R is normal, one obtains a generic bound for membership in the Frobenius closure. If d <= 2, then the bound for ideal membership in P can be computed from the known cases of the Froberg conjecture and yields explicit generic tight closure bounds.