Tight closure/Generic results/Section

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Is it more difficult to decide whether an element belongs to the tight closure of an ideal or to the ideal itself? We discuss one situation where this is easier for tight closure.

Suppose that we are in a graded situation of a given ring (or a given ring dimension) and have fixed a number (at least the ring dimension) of homogeneous generators and their degrees. Suppose that we want to know the degree bound for (tight closure or ideal) inclusion for generic choice of the ideal generators. Generic means that we write the coefficients of the generators as indeterminates and consider the situation over the (large) affine space corresponding to these indeterminates or over its function field. This problem is already interesting and difficult for the polynomial ring: Suppose we are in and want to study the generic inclusion bound for say generic polynomials all of degree . What is the minimal degree number such that

The answer is

This rests on the fact that the Fröberg conjecture is solved in dimension by D. Anick

(the Fröberg conjecture gives a precise description of the Hilbert function for an ideal in a polynomial ring which is generically generated. Here we only need to know in which degree the Hilbert function of the residue class ring becomes ).

The corresponding generic ideal inclusion bound for arbitrary graded rings depends heavily (already in the parameter case) on the ring itself. Surprisingly, the generic ideal inclusion bound for tight closure does not depend on the ring and is only slightly worse than the bound for the polynomial ring. The following theorem is due to Brenner and Fischbacher-Weitz.

Let and be natural numbers, . Let be a finite extension of standard-graded domains (a graded Noether normalization). Suppose that there exist homogeneous polynomials in with such that . Then

  1. holds in the generic point of the parameter space of homogeneous elements in of this degree type (the coefficients of the are taken as indeterminates).
  2. If is normal, then holds for (open) generic choice of homogeneous elements in of this degree type.

Suppose that we are in and that and . Then the generic degree bound for ideal inclusion in the polynomial ring is . Therefore by fact the generic degree bound for tight closure inclusion in a three-dimensional graded ring is .

Suppose that in the situation of fact. Then the generic elements are parameters. In the polynomial ring we have for parameters of degree the inclusion

because the graded Koszul resolution ends in and

So the theorem implies for a graded ring finite over that holds for generic elements. But by the graded Briançon-Skoda Theorem (see fact) this holds for parameters even without the generic assumption.