---
tag: XZS36C6226
kind: definition
name: Tight closure
msc: [13A35]
defines: KJ9MMP99G6
uses: [0670T59HGE, 0F6ES3BRYP, 096Z9MP2ME, S2D8823P8B, 0T3WVTD675, 0ZB2JG1M9C]
sources:
  - url: https://en.wikipedia.org/wiki/Tight_closure
---

## Definition

Let $p$ be a prime and let $R$ be a [[Noetherian ring|0670T59HGE]] of [[characteristic|0F6ES3BRYP]] $p$.
Let $R^\circ$ be the set of elements of $R$ that lie in no [[minimal prime ideal|S2D8823P8B]] of $R$.
For an [[ideal|096Z9MP2ME]] $I$ of $R$ and a power $q = p^e$ with $e \ge 0$, let $I^{[q]}$ be the ideal generated by the elements $a^q$ with $a \in I$.
An element $x \in R$ is in the *tight closure* $I^*$ of $I$ if there is $c \in R^\circ$ such that $c x^q \in I^{[q]}$ for all $q = p^e$ with $e$ large enough.

## Remarks

The ideal $I^{[q]}$ is the ideal generated by the image of $I$ under the $e$-th power of the [[Frobenius endomorphism|0T3WVTD675]].
If $I = (a_1, \ldots, a_m)$, then $I^{[q]} = (a_1^q, \ldots, a_m^q)$.
$I^*$ is an ideal, $I \subseteq I^*$, and $(I^*)^* = I^*$.
The ideal $I$ is *tightly closed* if $I^* = I$.

Example: let $k$ be a field of characteristic $p$, let $R = k[x, y]/(x^2)$, and let $I = (0)$.
The only minimal prime is $(x)$, so $1 \in R^\circ$, and $1 \cdot x^q = 0 \in I^{[q]}$ for $q \ge 2$.
So $x \in (0)^*$; more generally every [[nilpotent element|0ZB2JG1M9C]] lies in $(0)^*$.

Non-example: let $R = k[x]$ and $I = (x^2)$; then $x \notin I^*$.
Indeed $c x^q \in (x^{2q})$ means $x^q$ divides $c$, and if this holds for infinitely many $q$ then $c = 0$, which is not in $R^\circ$.