Tight closure

Summary

Tight closure is an operation on ideals of Noetherian rings of prime characteristic $p$, defined with the Frobenius endomorphism. It was introduced by Melvin Hochster and Craig Huneke. The tight closure $I^*$ of an ideal $I$ contains $I$ and is contained in the integral closure of $I$. In a regular ring, $I^* = I$. It gives short proofs of results such as the Hochster–Roberts theorem and the Briançon–Skoda theorem. It also leads to the classes of F-regular and F-rational rings.