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 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.
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?
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Karen Ellen Smith is an American mathematician, specializing in commutative algebra and algebraic geometry. She completed her bachelor's degree in mathematics at Princeton University before earning her PhD in mathematics at the University of Michigan in 1993. Currently she is the Keeler Professor of Mathematics at the University of Michigan. In addition to being a researcher in algebraic geometry and commutative algebra, Smith with others wrote the textbook An Invitation to Algebraic Geometry.
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