Tight closure

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In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by MelvinHochster and Craig Huneke  ( 1988 , 1990 ).

Let ${\displaystyle R}$ be a commutative noetherian ring containing a field of characteristic ${\displaystyle p>0}$. Hence ${\displaystyle p}$ is a prime number.

Let ${\displaystyle I}$ be an ideal of ${\displaystyle R}$. The tight closure of ${\displaystyle I}$, denoted by ${\displaystyle I^{*}}$, is another ideal of ${\displaystyle R}$ containing ${\displaystyle I}$. The ideal ${\displaystyle I^{*}}$ is defined as follows.

${\displaystyle z\in I^{*}}$ if and only if there exists a ${\displaystyle c\in R}$, where ${\displaystyle c}$ is not contained in any minimal prime ideal of ${\displaystyle R}$, such that ${\displaystyle cz^{p^{e}}\in I^{[p^{e}]}}$ for all ${\displaystyle e\gg 0}$. If ${\displaystyle R}$ is reduced, then one can instead consider all ${\displaystyle e>0}$.

Here ${\displaystyle I^{[p^{e}]}}$ is used to denote the ideal of ${\displaystyle R}$ generated by the ${\displaystyle p^{e}}$'th powers of elements of ${\displaystyle I}$, called the ${\displaystyle e}$th Frobenius power of ${\displaystyle I}$.

An ideal is called tightly closed if ${\displaystyle I=I^{*}}$. A ring in which all ideals are tightly closed is called weakly ${\displaystyle F}$-regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with localization, and so there is the additional notion of ${\displaystyle F}$-regular, which says that all ideals of the ring are still tightly closed in localizations of the ring.

Brenner & Monsky (2010) found a counterexample to the localization property of tight closure. However, there is still an open question of whether every weakly ${\displaystyle F}$-regular ring is ${\displaystyle F}$-regular. That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring is also tightly closed?

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References

• Brenner, Holger; Monsky, Paul (2010), "Tight closure does not commute with localization", Annals of Mathematics , Second Series, 171 (1): 571–588, arXiv:, doi:10.4007/annals.2010.171.571, ISSN   0003-486X, MR   2630050
• Hochster, Melvin; Huneke, Craig (1988), "Tightly closed ideals", Bulletin of the American Mathematical Society, New Series, 18 (1): 45–48, doi:, ISSN   0002-9904, MR   0919658
• Hochster, Melvin; Huneke, Craig (1990), "Tight closure, invariant theory, and the Briançon–Skoda theorem", Journal of the American Mathematical Society , 3 (1): 31–116, doi:10.2307/1990984, ISSN   0894-0347, JSTOR   1990984, MR   1017784