# Hilbert-Kunz theory/introduction/description

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

## Definition (Hilbert-Kunz function)

Let $K$ denote a field of positive characteristic $p$, let $K\subseteq R$ be a noetherian ring and let $I\subseteq R$ be an ideal which is primary to some maximal ideal. Then the Hilbert-Kunz function is the function
    $\varphi _{I}\colon \mathbb {N} \longrightarrow \mathbb {N} ,\,e\longmapsto \varphi _{I}(e)=\operatorname {length} \,(R/I^{[p^{e}]}),$
where $I^{[p^{e}]}$ is the extended ideal under the $e$-th iteration of the Frobenius homomorphism
    $R\longrightarrow R,\,f\longmapsto f^{p^{e}}.$

## Definition (Hilbert-Kunz multiplicity)

Let $K$ denote a field of positive characteristic $p$, let $K\subseteq R$ be a noetherian ring and let $I\subseteq R$ be an ideal which is primary to some maximal ideal of height $d$. Then the Hilbert-Kunz multiplicity of $I$ is the limit (if it exists)
    $\lim _{e\rightarrow \infty }{\frac {\operatorname {length} \,(R/I^{[p^{e}]})}{p^{ed}}}=\lim _{e\rightarrow \infty }{\frac {\varphi _{I}(e)}{p^{ed}}}\,.$
The Hilbert-Kunz multiplicity of the maximal ideal of a local noetherian ring $R$ is called the Hilbert-Kunz multiplicity of $R$.

## Theorem (Monsky)

Let $K$ denote a field of positive characteristic $p$, let $K\subseteq R$ be a noetherian ring and let $I\subseteq R$ be an ideal which is primary to some maximal ideal. Then the Hilbert-Kunz multiplicity $e_{HK}(I)$ exists and is a positive real number.
With the help of the Hilbert-Kunz invariant of a local noetherian ring one may characterize when $R$ is regular, as the following theorem shows (which was initiated by Kunz in 1969 but finally proven by Watanabe and Yoshida in 2000).

## Theorem (Watanabe, Yoshida)

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

## Theorem

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

## Theorem

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

Let $(R,{\mathfrak {m}})$ be an analytically unramified and formally equidimensional local noetherian ring of positive characteristic, let $I\subseteq R$ be an ${\mathfrak {m}}$-primary ideal. Let $f\in R$. Then
    $f\in I^{*}{\text{ if and only if }}e_{HK}((I,f))=e_{HK}(I).$
We are interested in the following three problems of the Hilbert-Kunz multiplicity.
* Is $e_{HK}(I)$ a rational number?
* In a relative situation, does there exist a limit for $\lim _{p\rightarrow \infty }e_{HK}(I_{p})$?
* 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 $A$ be a finitely generated $\mathbb {Z}$-domain ($A=\mathbb {Z}$ is a good example) and let $S$ be a noetherian $A$-algebra. This gives a family
    $\operatorname {Spec} {\left(S\right)}\longrightarrow \operatorname {Spec} {\left(A\right)}.$
For every maximal ideal ${\mathfrak {m}}$ of $A$ the residue class field $A/{\mathfrak {m}}$ is a finite field of some positive characteristic $p$, and the fiber ring $S_{\mathfrak {m}}=S\otimes _{A}A/{\mathfrak {m}}$ is a commutative ring of characteristic $p$. Over the prime ideal $(0)$ we get the $Q(A)$-algebra $S_{0}=S\otimes _{A}Q(A)$ of characteristic zero.
An ideal $I\subseteq S$ induces the extended ideal $I_{P}$, $P\in \operatorname {Spec} {\left(A\right)}$, in every fiber ring. If for all maximal ideals $P={\mathfrak {m}}$ these ideals are all primary to a maximal ideal in $S_{\mathfrak {m}}$, then we can compute the Hilbert-Kunz multiplicities
    $e_{HK}(I_{\mathfrak {m}})$
and can look what happens to these real numbers as the characteristic of $A/{\mathfrak {m}}$ tends to infinity. This limit, in case that it exists, should be an invariant of the generic fiber ring $S\otimes _{A}Q(A)$ and the ideal $I_{0}$ and should not depend on the relative family. There should also be an interpretation of this number which is independent of positive characteristic.