# Tight closure

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

Let be a commutative noetherian ring containing a field of characteristic . Hence is a prime number.

Let be an ideal of . The tight closure of , denoted by , is another ideal of containing . The ideal is defined as follows.

- if and only if there exists a , where is not contained in any minimal prime ideal of , such that for all . If is reduced, then one can instead consider all .

Here is used to denote the ideal of generated by the 'th powers of elements of , called the th Frobenius power of .

An ideal is called tightly closed if . A ring in which all ideals are tightly closed is called weakly -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 -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 -regular ring is -regular. That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring also tightly closed?

## References

- Brenner, Holger; Monsky, Paul (2010), "Tight closure does not commute with localization",
*Annals of Mathematics. Second Series*,**171**(1): 571–588, arXiv:0710.2913, doi:10.4007/annals.2010.171.571, ISSN 0003-486X, MR 2630050 - Hochster, Melvin; Huneke, Craig (1988), "Tightly closed ideals",
*American Mathematical Society. Bulletin. New Series*,**18**(1): 45–48, doi:10.1090/S0273-0979-1988-15592-9, ISSN 0002-9904, MR 919658 - 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, MR 1017784