Hilbert-Kunz Theorie/Graduierte Situation/Symmetrischer Zugang/en/Textabschnitt

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).