Tight closure/Generic results/Section

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 ${}P=K[X,Y,Z]$ and want to study the generic inclusion bound for say ${}n\geq 4$ generic polynomials ${}F_{1},\ldots ,F_{n}$ all of degree ${}a$. What is the minimal degree number ${}m$ such that

${}P_{\geq m}\subseteq {\left(F_{1},\ldots ,F_{n}\right)}\,.$

The answer is

$\left\lceil {\frac {1}{2(n-1)}}{\left(3-3n+2an+{\sqrt {1-2n+n^{2}+4a^{2}n}}\right)}\right\rceil .$

This rests on the fact that the Fröberg conjecture is solved in dimension ${}3$ 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 ${}0$).

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.

Theorem

Let ${}d\geq 1$ and ${}a_{1},\ldots ,a_{n}$ be natural numbers, ${}n\geq d+1$. Let ${}K[x_{0},x_{1},\ldots ,x_{d}]\subseteq R$ be a finite extension of standard-graded domains (a graded Noether normalization). Suppose that there exist ${}n$ homogeneous polynomials ${}g_{1},\ldots ,g_{n}$ in ${}P=K[x_{0},x_{1},\ldots ,x_{d}]$ with ${}\operatorname {deg} \,(g_{i})=a_{i}$ such that ${}P_{\geq m}\subseteq {\left(g_{1},\ldots ,g_{n}\right)}$. Then

1. ${}R_{m+d}\subseteq {\left(f_{1},\ldots ,f_{n}\right)}^{*}$ holds in the generic point of the parameter space of homogeneous elements ${}f_{1},\ldots ,f_{n}$ in ${}R$ of this degree type (the coefficients of the ${}f_{i}$ are taken as indeterminates).
2. If ${}R$ is normal, then ${}R_{m+d+1}\subseteq {\left(f_{1},\ldots ,f_{n}\right)}^{F}\subseteq {\left(f_{1},\ldots ,f_{n}\right)}^{*}$ holds for (open) generic choice of homogeneous elements ${}f_{1},\ldots ,f_{n}$ in ${}R$ of this degree type.

Example

Suppose that we are in ${}K[x,y,z]$ and that ${}n=4$ and ${}a=10$. Then the generic degree bound for ideal inclusion in the polynomial ring is ${}19$. Therefore by fact the generic degree bound for tight closure inclusion in a three-dimensional graded ring is ${}21$.

Example

Suppose that ${}n=d+1$ in the situation of fact. Then the generic elements ${}f_{1},\ldots ,f_{d+1}$ are parameters. In the polynomial ring ${}P=K[x_{0},x_{1},\ldots ,x_{d}]$ we have for parameters of degree ${}a_{1},\ldots ,a_{d+1}$ the inclusion

${}P_{\geq \sum _{i=0}^{d}a_{i}-d}\subseteq {\left(f_{1},\ldots ,f_{d+1}\right)}\,,$

because the graded Koszul resolution ends in ${}R(-\sum _{i=0}^{d}a_{i})$ and

$(H_{\mathfrak {m}}^{d+1}(P))_{k}=0{\text{ for }}k\geq -d.$

So the theorem implies for a graded ring ${}R$ finite over ${}P$ that ${}\subseteq (f_{1},\ldots ,f_{d+1})^{*}$ holds for generic elements. But by the graded Briançon-Skoda Theorem (see fact) this holds for parameters even without the generic assumption.