# Hilbert-Kunz theory/Introduction/Section

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

## Definition

Let ${\displaystyle {}K}$ denote a field of positive characteristic ${\displaystyle {}p}$, let ${\displaystyle {}K\subseteq R}$ be a noetherian ring and let ${\displaystyle {}I\subseteq R}$ be an ideal which is primary to some maximal ideal. Then the Hilbert-Kunz function is the function

${\displaystyle \varphi _{I}\colon \mathbb {N} \longrightarrow \mathbb {N} ,e\longmapsto \varphi _{I}(e)=\operatorname {length} \,(R/I^{[p^{e}]}),}$

where ${\displaystyle {}I^{[p^{e}]}}$ is the extended ideal under the ${\displaystyle {}e}$-th iteration of the Frobenius homomorphism

${\displaystyle R\longrightarrow R,f\longmapsto f^{p^{e}}.}$

## Definition

Let ${\displaystyle {}K}$ denote a field of positive characteristic ${\displaystyle {}p}$, let ${\displaystyle {}K\subseteq R}$ be a noetherian ring and let ${\displaystyle {}I\subseteq R}$ be an ideal which is primary to some maximal ideal of height ${\displaystyle {}d}$. Then the Hilbert-Kunz multiplicity of ${\displaystyle {}I}$ is the limit (if it exists)

${\displaystyle {}\lim _{e\rightarrow \infty }{\frac {\operatorname {length} \,(R/I^{[p^{e}]})}{p^{ed}}}=\lim _{e\rightarrow \infty }{\frac {\varphi _{I}(e)}{p^{ed}}}\,.}$

The Hilbert-Kunz multiplicity of the maximal ideal of a local noetherian ring ${\displaystyle {}R}$ is called the Hilbert-Kunz multiplicity of ${\displaystyle {}R}$.

## Theorem

Let ${\displaystyle {}K}$ denote a field of positive characteristic ${\displaystyle {}p}$, let ${\displaystyle {}K\subseteq R}$ be a noetherian ring and let ${\displaystyle {}I\subseteq R}$ be an ideal which is primary to some maximal ideal. Then the Hilbert-Kunz multiplicity ${\displaystyle {}e_{HK}(I)}$ exists and is a positive real number.

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

## Theorem

Let ${\displaystyle {}R}$ be a local noetherian ring of positive characteristic. Then the following hold.
1. The Hilbert-Kunz multiplicity of ${\displaystyle {}R}$ is ${\displaystyle {}e_{HK}(R)\geq 1}$.
2. If ${\displaystyle {}R}$ is unmixed, then ${\displaystyle {}e_{HK}(R)=1}$ if and only if ${\displaystyle {}R}$ is regular.

## Theorem

Let ${\displaystyle {}R}$ be a noetherian ring of positive characteristic of dimension one. Then the Hilbert-Kunz multiplicity of ${\displaystyle {}I\subseteq R}$ equals its Hilbert-Samuel multiplicity.

## Theorem

Let ${\displaystyle {}(R,{\mathfrak {m}})}$ be a regular local ring and let ${\displaystyle {}I}$ be an ${\displaystyle {}{\mathfrak {m}}}$-primary ideal. Then ${\displaystyle {}e_{HK}(I)=\operatorname {length} \,(R/I)}$.

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).

## Theorem

Let ${\displaystyle {}(R,{\mathfrak {m}})}$ be an analytically unramified and formally equidimensional local noetherian ring of positive characteristic, let ${\displaystyle {}I\subseteq R}$ be an ${\displaystyle {}{\mathfrak {m}}}$-primary ideal. Let ${\displaystyle {}f\in R}$. Then

${\displaystyle f\in I^{*}{\text{ if and only if }}e_{HK}((I,f))=e_{HK}(I).}$

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

1. Is ${\displaystyle {}e_{HK}(I)}$ a rational number?
2. In a relative situation, does there exist a limit for ${\displaystyle {}\lim _{p\rightarrow \infty }e_{HK}(I_{p})_{p}}$?
3. 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 ${\displaystyle {}A}$ be a finitely generated ${\displaystyle {}\mathbb {Z} }$-domain (${\displaystyle {}A=\mathbb {Z} }$ is a good example) and let ${\displaystyle {}S}$ be a noetherian ${\displaystyle {}A}$-algebra. This gives a family

${\displaystyle \operatorname {Spec} {\left(S\right)}\longrightarrow \operatorname {Spec} {\left(A\right)}.}$

For every maximal ideal ${\displaystyle {}{\mathfrak {m}}}$ of ${\displaystyle {}A}$ the residue class field ${\displaystyle {}A/{\mathfrak {m}}}$ is a finite field of some positive characteristic ${\displaystyle {}p}$, and the fiber ring ${\displaystyle {}S_{\mathfrak {m}}=S\otimes _{A}A/{\mathfrak {m}}}$ is a commutative ring of characteristic ${\displaystyle {}p}$. Over the prime ideal ${\displaystyle {}(0)}$ we get the ${\displaystyle {}Q(A)}$-algebra ${\displaystyle {}S_{0}=S\otimes _{A}Q(A)}$ of characteristic zero.

An ideal ${\displaystyle {}I\subseteq S}$ induces the extended ideal ${\displaystyle {}I_{P}}$, ${\displaystyle {}P\in \operatorname {Spec} {\left(A\right)}}$, in every fiber ring. If for all maximal ideals ${\displaystyle {}P={\mathfrak {m}}}$ these ideals are all primary to a maximal ideal in ${\displaystyle {}S_{\mathfrak {m}}}$, then we can compute the Hilbert-Kunz multiplicities

${\displaystyle e_{HK}(I_{\mathfrak {m}})}$

and can look what happens to these real numbers as the characteristic of ${\displaystyle {}A/{\mathfrak {m}}}$ tends to infinity. This limit, in case that it exists, should be an invariant of the generic fiber ring ${\displaystyle {}S\otimes _{A}Q(A)}$ and the ideal ${\displaystyle {}I_{0}}$ and should not depend on the relative family. There should also be an interpretation of this number which is independent of positive characteristic.