Hilbert-Kunz theory/Introduction/Section

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

Definition

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

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

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

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 multiplicity.

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

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.

1. Is ${}e_{HK}(I)$ a rational number?
2. In a relative situation, does there exist a limit for ${}\lim _{p\rightarrow \infty }e_{HK}(I_{p})_{p}$?
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 ${}A$ 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.