Math 2.0

Commutative algebra Polynomial rings and invariants Theorem

Kunz's theorem

Builds on Noetherian ringCharacteristic of a ringFrobenius endomorphismFlat moduleLocalizationPrime idealRegular local ring

Statement

Let pp be a prime and let RR be a Noetherian ring of characteristic pp. Let F ⁣:R→RF \colon R \to R, F(r)=rpF(r) = r^p, be the Frobenius endomorphism, and view RR as an RR-module through FF. Then the following are equivalent:

  1. RR is a flat RR-module through FF.
  2. RR is regular, that is, for every prime ideal 𝔭\mathfrak p of RR, the localization R𝔭R_{\mathfrak p} is a regular local ring.

Remarks

Both conditions are local, so one may assume that RR is local with maximal ideal 𝔪\mathfrak m. If RR is regular, FF is a local homomorphism from a regular local ring to a Cohen–Macaulay ring with dim⁡R/𝔪[p]=0\dim R/\mathfrak m^{[p]} = 0, and such a map is flat ("miracle flatness"). The converse is the deeper direction, and it was the main new point of Kunz's work.

Example: for R=𝔽p[x]R = \F_p[x], the image of FF is 𝔽p[xp]\F_p[x^p]. The ring RR is a free module over it with basis 1,x,…,xp−11, x, \ldots, x^{p-1}, so FF is flat, as the theorem predicts for this regular ring. For R=𝔽p[x,y]/(xy)R = \F_p[x, y]/(xy), which is not regular at (x,y)(x, y), the Frobenius map is not flat.

Going deeper

For a student

The theorem is the starting point of the use of Frobenius to study singularities in prime characteristic. It shows, for instance, that in a regular ring every ideal is tightly closed: from cxq∈I[q]c x^q \in I^{[q]} and flatness one gets c∈(I[q]:xq)=(I:x)[q]c \in (I^{[q]} : x^q) = (I : x)^{[q]}, which forces x∈Ix \in I. Weakening flatness of Frobenius leads to the classes of F-regular, F-rational, F-pure and F-injective rings, see tight closure.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Characteristic of a ring, concept, no evidence levelCharacteristic of a r…conceptFlat module, concept, no evidence levelFlat moduleconceptFrobenius endomorphism, concept, no evidence levelFrobenius endomorphismconceptLocalization, concept, no evidence levelLocalizationconceptNoetherian ring, concept, no evidence levelNoetherian ringconceptPrime ideal, concept, no evidence levelPrime idealconceptRegular local ring, concept, no evidence levelRegular local ringconceptKunz's theorem, statement, no evidence levelKunz's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Characteristic of a ring concept
  2. Flat module concept
  3. Frobenius endomorphism concept
  4. Localization concept
  5. Noetherian ring concept
  6. Prime ideal concept
  7. Regular local ring concept
  8. Kunz's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Frobenius_endomorphism
About this page
TagFRCY5BQF6N
URIm2://FRCY5BQF6N
Kindstatement, theorem
WritingW0 stub
MSC13A35, 13H05
StatusThis copy of the site has no build data. Run m2 status FRCY5BQF6N.
AboutFrobenius endomorphism, Regular local ring
UsesNoetherian ring, Characteristic of a ring, Frobenius endomorphism, Flat module, Localization, Prime ideal, Regular local ring
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.