# Hilbert-Kunz theory/Introduction/Section

In 1969, Kunz considered first the following function and the corresponding limit.

Let denote a field of positive characteristic , let
be a noetherian ring and let
be an ideal which is primary to some maximal ideal. Then the *Hilbert-Kunz function* is the function

where is the extended ideal under the -th iteration of the Frobenius homomorphism

Let denote a field of positive characteristic , let
be a noetherian ring and let
be an ideal which is primary to some maximal ideal of height . Then the *Hilbert-Kunz multiplicity* of is the limit
(if it exists)

The Hilbert-Kunz multiplicity of the maximal ideal of a local noetherian ring is called the *Hilbert-Kunz multiplicity* of .

Let denote a field of positive characteristic , let be a noetherian ring and let be an ideal which is primary to some maximal ideal. Then the Hilbert-Kunz multiplicity exists and is a positive real number.

With the help of the Hilbert-Kunz invariant of a local noetherian ring one may characterize when is regular, as the following theorem shows
(which was initiated by Kunz in 1969 but finally proven by Watanabe and Yoshida in 2000).

- The Hilbert-Kunz multiplicity of is .
- If is unmixed, then if and only if is regular.

Let be a noetherian ring of positive characteristic of dimension one. Then the Hilbert-Kunz multiplicity of equals its Hilbert-Samuel multiplicity.

Let be a regular local ring and let be an -primary ideal. Then .

There is a direct relation between Hilbert-Kunz multiplicity and tight closure
(and the test ideal of tight closure theory is related to the multiplier ideal of the ideal).

Let be an analytically unramified and formally equidimensional local noetherian ring of positive characteristic, let be an -primary ideal. Let . Then

We are interested in the following three problems of the Hilbert-Kunz multiplicity.

- Is a rational number?
- In a relative situation, does there exist a limit for ?
- Is there a direct interpretation of the Hilbert-Kunz multiplicity in characteristic zero (which coincides with the limit in the relative situation, if this limit exists)?

We explain the relative situation: Let be a finitely generated -domain ( is a good example) and let be a noetherian -algebra. This gives a family

For every maximal ideal of the residue class field is a finite field of some positive characteristic , and the fiber ring is a commutative ring of characteristic . Over the prime ideal we get the -algebra of characteristic zero.

An ideal induces the extended ideal , , in every fiber ring. If for all maximal ideals these ideals are all primary to a maximal ideal in , then we can compute the Hilbert-Kunz multiplicities

and can look what happens to these real numbers as the characteristic of tends to infinity. This limit, in case that it exists, should be an invariant of the generic fiber ring and the ideal and should not depend on the relative family. There should also be an interpretation of this number which is independent of positive characteristic.