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**Tight closure does not commute with localization.**
*(English)*
Zbl 1239.13008

Tight closure is an operation on ideals in commutative rings of characteristic \(p>0\) defined by M. Hochster and C. L. Huneke in the mid-1980s [J. Am. Math. Soc. 3, No. 1, 31–116 (1990; Zbl 0701.13002)]. It is defined as follows: suppose we are given an ideal \(I=(f_1,\dots,f_n)\) in a ring \(R\). The tight closure of \(I\), denoted by \(I^*\), is the set of elements \(z \in R\) for which there exists some element \(c \in R\), not contained in any minimal prime of \(R\), such that \(cz^{p^e} \in (f^{p^e}_1,\dots,f^{p^e}_n)\) for all \(e\gg0\). While tight closure has had numerous interesting applications and has been studied by dozens of authors, perhaps the most fundamental question was left unanswered:

Question: Does the formation of tight closure commute with localization?

For many years it was hoped (and perhaps even expected) that this question would have a positive answer. However, this breakthrough paper shows that the answer to this question is no. As a consequence, tight closure is not the same as plus closure (a closure operation known to commute with localization). Recall that the plus closure \(I^+\) of an ideal \(I\) is defined by extending \(I\) to the integral closure of \(R\) in the algebraic closure of its field of fractions, and then contracting back to \(R\). It had been previously shown by K. E. Smith [Invent. Math. 115, No. 1, 41–60 (1994; Zbl 0820.13007)] that \(I^+=I^*\) when \(I\) is a parameter ideal.

The first author has extensively studied tight closure and plus closure in two-dimensional graded rings [J. Algebra 265, No. 1, 45–78 (2003; Zbl 1099.13010)]. Considering 1-parameter families of such rings leads to the study of three-dimensional rings, where the example was constructed. On the other hand, the explicit example has its roots in the study of Hilbert-Kunz multiplicity [J. Algebra 208, No. 1, 343–358 (1998; Zbl 0932.13010)] by the second author. Hilbert-Kunz multiplicity is an invariant of ideals in rings of characteristic \(p>0\) which is intimately related to tight closure see M. Hochster and C. L. Huneke, [J. Am. Math. Soc. 3, No. 1, 31–116 (1990; Zbl 0701.13002)].

The authors point out that it may well still be possible that the formation of tight closure commutes with localization in certain geometric settings. For example, it is unknown whether tight closure commutes with the inversion of a single element, or whether tight closure commutes with localization for ideals in rings of finite type over a finite field.

Question: Does the formation of tight closure commute with localization?

For many years it was hoped (and perhaps even expected) that this question would have a positive answer. However, this breakthrough paper shows that the answer to this question is no. As a consequence, tight closure is not the same as plus closure (a closure operation known to commute with localization). Recall that the plus closure \(I^+\) of an ideal \(I\) is defined by extending \(I\) to the integral closure of \(R\) in the algebraic closure of its field of fractions, and then contracting back to \(R\). It had been previously shown by K. E. Smith [Invent. Math. 115, No. 1, 41–60 (1994; Zbl 0820.13007)] that \(I^+=I^*\) when \(I\) is a parameter ideal.

The first author has extensively studied tight closure and plus closure in two-dimensional graded rings [J. Algebra 265, No. 1, 45–78 (2003; Zbl 1099.13010)]. Considering 1-parameter families of such rings leads to the study of three-dimensional rings, where the example was constructed. On the other hand, the explicit example has its roots in the study of Hilbert-Kunz multiplicity [J. Algebra 208, No. 1, 343–358 (1998; Zbl 0932.13010)] by the second author. Hilbert-Kunz multiplicity is an invariant of ideals in rings of characteristic \(p>0\) which is intimately related to tight closure see M. Hochster and C. L. Huneke, [J. Am. Math. Soc. 3, No. 1, 31–116 (1990; Zbl 0701.13002)].

The authors point out that it may well still be possible that the formation of tight closure commutes with localization in certain geometric settings. For example, it is unknown whether tight closure commutes with the inversion of a single element, or whether tight closure commutes with localization for ideals in rings of finite type over a finite field.

### MSC:

13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |

14H60 | Vector bundles on curves and their moduli |

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\textit{H. Brenner} and \textit{P. Monsky}, Ann. Math. (2) 171, No. 1, 571--588 (2010; Zbl 1239.13008)

### References:

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[18] | P. Monsky, ”Hilbert-Kunz functions in a family: Point-\(S_4\) quartics,” J. Algebra, vol. 208, iss. 1, pp. 343-358, 1998. · Zbl 0932.13010 · doi:10.1006/jabr.1998.7500 |

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