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 is also tightly closed?
- 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 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