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 Mel Hochster and Craig Huneke in the 1980s.

Let $R$ be a commutative noetherian ring containing a field of characteristic $p > 0$ . Hence $p$ is a prime.

Let $I$ be an ideal of $R$ . The tight closure of $I$ , denoted by $I *$ , is another ideal of $R$ containing $I$ . It is defined as follows.

$z \in I^*$ if and only if there exists a $c \in R$ , where

c

is not contained in any minimal prime ideal of

R

, such that $c z^{p^e} \in I^{[p^e]}$ for all $e \gg 0$ .

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

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