Benutzer:Holger Brenner/Talk in Luminy January 2011

The Hilbert-Kunz multiplicity - and what it might be in characteristic zero

Hilbert-Kunz function and Hilbert-Kunz multiplicity

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.

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

We want to discuss some situations where the answer to these problems is known. A typical feature of Hilbert-Kunz theory is that in special situations there is sometimes a special interpretation of the Hilbert-Kunz multiplicity, and then a positive answer to all questions above follow from this. If there is no such interpretation, then usually nothing is known.

Hilbert-Kunz theory for graded rings over curves

We will restrict now to the two-dimensional homogeneous case in order to work on the corresponding projective curve.

Let ${}R$ be a two-dimensional standard-graded normal domain over an algebraically closed field ${}K$. Let

${}C=\operatorname {Proj} {\left(R\right)}\,$

be the corresponding smooth projective curve and let

${}I={\left(f_{1},\ldots ,f_{n}\right)}\,$

be an ${}R_{+}$-primary homogeneous ideal with generators of degrees ${}d_{1},\ldots ,d_{n}$. Then we get on ${}C$ the short exact sequence

$0\longrightarrow \operatorname {Syz} \,(f_{1},\ldots ,f_{n})(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0.$

Here ${}\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)$ is a vector bundle, called the syzygy bundle, of rank ${}n-1$ and of degree

$((n-1)m-\sum _{i=1}^{n}d_{i})\operatorname {deg} \,(C).$

Our approach to the computation of the Hilbert-Kunz multiplicity is by using the presenting sequence

$0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}\longrightarrow 0$

and twists of its ${}e$-th Frobenius pull-backs, that is

$0\longrightarrow \operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-qd_{i}){\stackrel {f_{1}^{q},\ldots ,f_{n}^{q}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0$

(where ${}q=p^{e}$), and to relate the asymptotic behavior of

${}\operatorname {length} \,(R/I^{[q]})=\operatorname {dim} _{K}\,(R/I^{[q]})=\sum _{m=0}^{\infty }\operatorname {dim} _{K}\,(R/I^{[q]})_{m}\,$

to the asymptotic behavior of the global sections of the Frobenius pull-backs

${}(F^{e*}(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})(m)=\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m)\,.$

What we want to compute is just the cokernel of the complex of global sections of the above sequence, namely

$\operatorname {dim} _{K}\,(R/I^{[q]})_{m}=h^{0}(C,{\mathcal {O}}_{C}(m))-\sum _{i=1}^{n}h^{0}(C,{\mathcal {O}}_{C}(m-qd_{i}))+h^{0}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))\,.$

The summation over ${}m$ is finite (but the range depends on ${}q$), and the terms

${}h^{0}(C,{\mathcal {O}}_{C}(m))=\operatorname {dim} _{K}\,\Gamma (C,{\mathcal {O}}_{C}(m))=\operatorname {dim} _{K}\,R_{m}\,$

are easy to control, so we have to understand the behavior of the global syzygies

$H^{0}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(m))$

for all ${}q$ and ${}m$, at least asymptotically. This is a Frobenius-Riemann-Roch problem (so far this works for all normal standard-graded domains).

The strategy for this is to use Riemann-Roch to get a formula for ${}H^{0}-H^{1}$ and then use semistability properties to show that ${}H^{0}$ or ${}H^{1}$ are ${}0$ in certain ranges.

We need the concept of (strong) semistability.

Definition (semistable and strongly semistable)

Let ${}{\mathcal {S}}$ be a vector bundle on a smooth projective curve ${}C$. It is called semistable, if ${}\mu ({\mathcal {T}})={\frac {\deg({\mathcal {T}})}{\operatorname {rk} ({\mathcal {T}})}}\leq {\frac {\deg({\mathcal {S}})}{\operatorname {rk} ({\mathcal {S}})}}=\mu ({\mathcal {S}})$ for all subbundles ${}{\mathcal {T}}$.

Suppose that the base field has positive characteristic ${}p>0$. Then ${}{\mathcal {S}}$ is called strongly semistable, if all (absolute) Frobenius pull-backs ${}F^{e*}({\mathcal {S}})$ are semistable.

The rational number ${}\mu ({\mathcal {S}})={\frac {\operatorname {deg} ({\mathcal {S}})}{\operatorname {rk} ({\mathcal {S}})}}$ is called the slope of a vector bundle.

Definition (Harder-Narasimhan filtration)

Let ${}{\mathcal {S}}$ be a vector bundle on a smooth projective curve ${}C$ over an algebraically closed field ${}K$. Then the (uniquely determined) filtration

${}0={\mathcal {S}}_{0}\subset {\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}={\mathcal {S}}\,$

of subbundles such that all quotient bundles ${}{\mathcal {S}}_{k}/{\mathcal {S}}_{k-1}$ are semistable with decreasing slopes ${}\mu _{k}=\mu ({\mathcal {S}}_{k}/{\mathcal {S}}_{k-1})$, is called the Harder-Narasimhan filtration of ${}{\mathcal {S}}$.

Theorem (Langer)

Let ${}C$ denote a smooth projective curve over an algebraically closed field of positive characteristic ${}p$, and let ${}{\mathcal {S}}$ be a vector bundle on ${}C$. Then there exists a natural number ${}e\in \mathbb {N}$ such that the Harder-Narasimhan filtration of the ${}e$-th Frobenius pull-back ${}F^{e*}({\mathcal {S}})$, say

${}0={\mathcal {S}}_{0}\subset {\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}=F^{e*}({\mathcal {S}})\,$

has the property that the quotients ${}{\mathcal {S}}_{k}/{\mathcal {S}}_{k-1}$ are strongly semistable.

An immediate consequence of this is that the Harder-Narasimhan filtration of all higher Frobenius pull-backs are just the pull-backs of this filtration. With these filtration we can at least Frobenius-asymptotically control the global sections of the pull-backs and hence also the Hilbert-Kunz multiplicity. This implies the following theorem.

Theorem (Brenner, Trivedi)

Let ${}R$ be a two-dimensional standard-graded normal domain over an algebraically closed field of positive characteristic. Let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be a homogeneous ${}R_{+}$-primary ideal with homogeneous generators of degree ${}d_{i}$. Let ${}{\mathcal {S}}=\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}$ be the syzygy bundle on ${}C=\operatorname {Proj} {\left(R\right)}$ and suppose that the Harder-Narasimhan filtration of ${}F^{e*}({\mathcal {S}})$ is strong, and let ${}\mu _{k}$, ${}k=1,\ldots ,t$, be the corresponding slopes. We set ${}\nu _{k}={\frac {-\mu _{k}}{\operatorname {deg} \,(C)p^{e}}}$ and ${}r_{k}=\operatorname {rk} \,({\mathcal {S}}_{k}/{\mathcal {S}}_{k-1})$. Then the Hilbert-Kunz multiplicity of ${}I$ is

${}e_{HK}(I)={\frac {\operatorname {deg} \,(C)}{2}}{\left(\sum _{k=1}^{t}r_{k}\nu _{k}^{2}-\sum _{i=1}^{n}d_{i}^{2}\right)}\,.$

In particular, it is a rational number.

Corollary (Brenner, Trivedi)

Let ${}R=K[x,y,z]/(H)$ be a normal homogeneous hypersurface domain of dimension two and degree ${}\delta$ over an algebraically closed field of positive characteristic. Then there exists a rational number ${}\nu _{2}$, ${}{\frac {3}{2}}\leq \nu _{2}\leq 2$, such that the Hilbert-Kunz multiplicity of ${}R$ is

${}e_{HK}(R)=\delta {\left(\nu _{2}^{2}-3\nu _{2}+3\right)}\,.$

Relative curves

Theorem (Trivedi)

Let

$C\longrightarrow \operatorname {Spec} {\left(\mathbb {Z} _{g}\right)}$

be a smooth projective relative curve and let ${}{\mathcal {T}}$ be a vector bundle over ${}C$. For the generic fiber ${}C_{0}$ let

${}\mu _{HK}({\mathcal {T}}_{0})=\sum _{\ell }r_{\ell }\mu _{\ell }^{2}\,,$

where the ${}\mu _{\ell }$ are the slopes (and ${}r_{\ell }$ the ranks) in the Harder-Narasimhan filtration of ${}{\mathcal {T}}_{0}$, and for every prime number let

${}{\bar {\mu }}_{HK}({\mathcal {T}}_{p})={\frac {\sum _{k}r_{k}\mu _{k}^{2}}{p^{e}}}\,,$

where the ${}\mu _{k}$ are the slopes in the strong Harder-Narasimhan filtration of ${}F^{e*}({\mathcal {T}}_{p})$ on the fiber ${}C_{p}$. Then

${}\lim _{p\rightarrow \infty }{\bar {\mu }}_{HK}({\mathcal {T}}_{p})=\mu _{HK}({\mathcal {T}}_{0})\,$

In charcteristic zero and a given homogeneous ideal, we use the slopes from the Harder-Narasimhan filtration of the syzygy bundle to define the Hilbert-Kunz multiplicity by

${}e_{HK}^{0}(I)={\frac {1}{2\operatorname {deg} \,C}}{\left(\mu _{HK}(\operatorname {Syz} )-(\operatorname {deg} \,C)^{2}\sum _{i=1}^{n}d_{i}^{2}\right)}\,.$

In the relative situation, an ideal gives a syzygy bundle over the relative curve and there we can compare the slopes coming from a strong Harder-Narasimhan filtration in the fibers to get the following Corollary.

Corollary (Trivedi)

Let ${}\mathbb {Z} \subseteq S$ be a finitely generated graded domain of relative dimension two with normal fibers and let ${}I\subseteq S$ be a homogeneous ideal such that ${}I_{p}$ is an ${}(S\otimes _{\mathbb {Z} }\mathbb {Z} /(p))_{+}$-primary ideal for all prime reductions. Then the limit

$\lim _{p\rightarrow \infty }e_{HK}(I_{p})$

exists and equals the Hilbert-Kunz multiplicity ${}e_{HK}^{0}(I_{0})$ in characteristic zero.

It is also known by a result of Brenner, Li, Miller, that it is enough to compute in every positive characteristic ${}p$ just the first Frobenius pull-back (not all Frobenius powers) and then let ${}p$ go to infinity to get the same limit.

Example

The Fermat-quartic ${}x^{4}+y^{4}+z^{4}=0$ is the easiest example where the Hilbert-Kunz multiplicity of the maximal ideal ${}(x,y,z)$ fluctuates with the characteristic. We have

${}e_{HK}(R_{p})={\begin{cases}3{\text{ for }}p=\pm 1\,\operatorname {mod} \,8\\3+{\frac {1}{p^{2}}}{\text{ for }}p=\pm 3\,\operatorname {mod} \,8\,.\end{cases}}\,$

The limit is of course ${}3$, which corresponds to the fact that the syzygy bundle is semistable in characteristic zero. The syzygy bundle is semistable for all prime characteristics ${}p\geq 3$, but not strongly semistable for the prime numbers ${}p=\pm 3\,\operatorname {mod} \,8$.

The symmetric approach to Hilbert-Kunz theory

We discuss briefly an approach of Brenner and Fischbacher-Weitz to replace the Frobenius asymptotic (which is only available in positive characteristic and where, in a relative situation, different Frobenius homomorphisms occur) by the symmetric asymptotic. This means that instead of global sections of Frobenius pull-backs of the syzygy bundle ${}\operatorname {Syz}$ (on a projective variety) we look at the global sections of the symmetric powers

$S^{k}(\operatorname {Syz} ).$

This approach works at the moment only in graded ring dimension two, the appropriate generalization to higher dimensions is not clear right now. In dimension two however, this approach gives even a symmetric Hilbert-Kunz function, which assigns to every natural number ${}k$ a rational number, which is „close“ to ${}{\frac {\varphi _{I_{p}}(e)}{q^{2}}}$ if ${}q$ is a power of ${}p$. It is defined in the following way. We start again with the presenting sequence

$0\longrightarrow \operatorname {Syz} \longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(-d_{i})\longrightarrow {\mathcal {O}}_{C}\longrightarrow 0.$

This sequence yields

$0\longrightarrow S^{q}(\operatorname {Syz} )\longrightarrow S^{q}(\bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(-d_{i}))\longrightarrow S^{q-1}(\bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(-d_{i}))\longrightarrow 0.$

The global mapping of suitable twists of this squence is

$\psi _{q,m}\colon H^{0}(C,(S^{q}(\bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(-d_{i})))(m))\longrightarrow H^{0}(C,(S^{q-1}(\bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(-d_{i})))(m)).$

These mapping can be computed explicitly. We make the following definition.

Definition

For a homogeneous ${}R_{+}$-primary ideal ${}I$ in a two-dimensional standard-graded normal domain ${}R$ we define

${}\varphi _{I}^{S}(q)=\sum _{m=0}^{\infty }\operatorname {dim} _{K}\,(\operatorname {coker} \,\psi _{q,m})\,$

and call this the symmetric Hilbert-Kunz function.

The ranks of the symmetric powers and hence the global sections grow fast, we have to look at

${\frac {\sum _{m=0}^{\infty }\operatorname {dim} _{K}\,(\operatorname {coker} \,\psi _{q,m})}{\binom {q+n-1}{n}}}.$

Definition

The symmetric Hilbert-Kunz multiplicity is defined by

${}e_{HK}^{S}(I)=\lim _{q\rightarrow \infty }{\frac {\sum _{m=0}^{\infty }\operatorname {dim} _{K}\,(\operatorname {coker} \,\psi _{q,m})}{\binom {q+n-1}{n}}}\,.$

For the computation on the Fermat-Quartic in characteristic zero see here.

In characteristic zero we have the following theorem.

Theorem (Brenner, Fischbacher-Weitz)

Let ${}K$ be an algebraically closed field of characteristic zero and let ${}R$ be a normal standard-graded two-dimensional ${}K$-domain. Let ${}I$ denote a homogeneous ${}R_{+}$-primary ideal. Then

$e_{HK}^{S}(I)=e_{HK}^{0}(I).$

This rests upon the fact that from the asymptotic behaviour of the global sections of the symmetric powers of a bundle we can read of the slopes in its Harder-Narasimhan filtration (this is not possible from the tensor powers).

Hilbert-Kunz theory and invariant theory

Let ${}G$ be a finite group acting linearly on a polynomial ring ${}B=K[x_{1},\ldots ,x_{n}]$ with invariant ring ${}A=B^{G}$. This is a positively graded ${}K$-algebra with irrelevant ideal ${}A_{+}$ consisting of all invariant polynomials of positive degree. The extended ideal ${}{\mathfrak {h}}=A_{+}B$ is called the Hilbert ideal. The residue class ring

${}B_{G}=B/A_{+}B\,$

is called the ring of coinvariants.

We are interested in the Hilbert-Kunz multiplicity of ${}A$ and of its localization at the irrelevant ideal. A result of Watanabe and Yoshida implies the following observation. It uses the fact that for regular rings the Hilbert-Kunz multiplicity of an ideal is just the colength.

Lemma

Let ${}G$ be a finite group acting linearly on a polynomial ring ${}B=K[x_{1},\ldots ,x_{n}]$ with invariant ring ${}A=B^{G}$ and let ${}{\mathfrak {h}}=A_{+}B$ be the Hilbert ideal in ${}B$. Let ${}R$ be the localization of ${}A$ at the irrelevant ideal. If ${}K$ has positive characteristic, then

${}e_{HK}(R)={\frac {\operatorname {dim} _{K}\,(B/{\mathfrak {h}})}{\#\left(G\right)}}\,.$

In particular, this is a rational number, and this quotient depends only on the invariant ring, not on its representation.

With this observation we can give a Hilbert-Kunz proof of the following theorem of invariant theory (which was proved in positive characteristic by Larry Smith).

Corollary

Let ${}G$ be a finite group acting linearly on a polynomial ring ${}B=K[x_{1},\ldots ,x_{n}]$ with invariant ring ${}A=B^{G}$ and let ${}{\mathfrak {h}}=A_{+}B$ be the Hilbert ideal. If ${}K$ has positive characteristic, then the invariant ring is a polynomial ring if and only if we have

${}\operatorname {dim} _{K}\,B/{\mathfrak {h}}={\#\left(G\right)}\,.$

Brieskorn singularities

Remark  

We consider the Brieskorn singularities

${}S=\mathbb {Z} /(p)[x_{1},\ldots ,x_{s}]/{\left(x_{1}^{d_{1}}+x_{2}^{d_{2}}+\cdots +x_{s}^{d_{s}}\right)}\,$

together with the maximal ideal. Also in this case it is known (Monsky-Han) that the Hilbert-Kunz multiplicities are rational and that they converge to a rational number for ${}p\mapsto \infty$. There exists an explicit but not easy description for this number. If all exponents are ${}2$, then the limit is

$1+{\text{coefficient of }}z^{s-1}{\text{ in the power series expansion of }}\operatorname {sec} \,z+\operatorname {tan} \,z.$