---
tag: KJ9MMP99G6
kind: concept
name: Tight closure
msc: [13A35]
sources:
  - url: https://en.wikipedia.org/wiki/Tight_closure
same_as:
  wikidata: Q7801630
  wikipedia: Tight closure
---

## Summary

Tight closure is an operation on ideals of Noetherian rings of prime characteristic $p$, defined with the [[Frobenius endomorphism|0T3WVTD675]].
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|56107GMFYV]] and the Briançon–Skoda theorem.
It also leads to the classes of F-regular and F-rational rings.

## Explanations

### L1 human

In characteristic $p$, raising to the $p$-th power is a ring homomorphism, and one can raise elements to the powers $q = p, p^2, p^3, \ldots$.
An element $x$ is in the tight closure of $I$ when its powers $x^q$ almost lie in the ideal generated by the $q$-th powers of elements of $I$.
"Almost" means: up to one fixed multiplier $c$ that is not too degenerate.
Elements of $I$ are in it, and so are elements that differ from $I$ by something "infinitely small", such as a nilpotent element.
In a polynomial ring, nothing else is added: every ideal is tightly closed.