Vector bundles and tight closure (Triest 2023)/Lecture 1

Hilbert-Kunz theory

In 1969, Kunz was the first to consider 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 ${}R$. The existence of Hilbert-Kunz multiplicity was proven by Monsky.

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 multiplicity 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.

Vector bundles

We will have a look at Hilbert-Kunz theory and tight closure (to be introduced in the next lecture) from the viewpoint of vector bundles. To motivate this concept, which exists in algebraic geometry, differential geometry, topology, mathematical physics, we go back to linear algebra. Let ${}K$ be a field. We consider a system of linear homogeneous equations over ${}K$,

${}f_{11}t_{1}+\cdots +f_{1n}t_{n}=0\,,$

${}f_{21}t_{1}+\cdots +f_{2n}t_{n}=0\,,$

$\vdots$

${}f_{m1}t_{1}+\cdots +f_{mn}t_{n}=0\,,$

where the ${}f_{ij}$ are elements in ${}K$. The solution set to this system of homogeneous equations is a vector space ${}V$ over ${}K$ (a linear subspace of ${}K^{n}$), its dimension is ${}n-\operatorname {rk} (A)$, where

${}A={\left(f_{ij}\right)}_{ij}\,$

is the matrix given by these elements. Suppose now that ${}X$ is a geometric object (a topological space, a manifold, a variety, a scheme, the spectrum of a ring) and that instead of elements in the field ${}K$ we have functions

$f_{ij}\colon X\longrightarrow K$

on ${}X$ (which are continuous, or differentiable, or algebraic). We form the matrix of functions ${}A={\left(f_{ij}\right)}_{ij}$, which yields for every point ${}P\in X$ a matrix ${}A(P)$ over ${}K$. Then we get from these data the space

${}V={\left\{(P;t_{1},\ldots ,t_{n})\mid A(P){\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}=0\right\}}\subseteq X\times K^{n}\,$

together with the projection to ${}X$. For a fixed point ${}P\in X$, the fiber ${}V_{P}$ of ${}V$ over ${}P$ is the solution space in ${}K^{n}$ to the corresponding system of homogeneous linear equations given by inserting ${}P$ into ${}f_{ij}$. In particular, all fibers of the map

$V\longrightarrow X,$

are vector spaces (maybe of non-constant dimension). These vector space structures yield an addition

$V\times _{X}V\longrightarrow V,(P;s_{1},\ldots ,s_{n};t_{1},\ldots ,t_{n})\longmapsto (P;s_{1}+t_{1},\ldots ,s_{n}+t_{n}).$

Here, ${}V\times _{X}V$ is the fiber product of ${}V\rightarrow X$ with itself, only points in the same fiber can be added. The mapping

$X\longrightarrow V,P\longmapsto (P;0,\ldots ,0),$

is called the zero-section. If we have just one equation with functions ${}f_{1},\ldots ,f_{n}$ and if ${}U\subseteq X$ denotes the open subset where not all ${}f_{i}$ vanish, then we get a short exact sequence

$0\longrightarrow V{|}_{U}\longrightarrow U\times K^{n}\longrightarrow U\times K\longrightarrow 0.$

If ${}D(f_{i})\subseteq U$ denotes the locus where ${}f_{i}$ does not vanish, then we get a linear isomorphism

$V{|}_{D(f_{i})}\longrightarrow D(f_{i})\times K^{n-1},\left(t_{1},\,\ldots ,\,t_{n}\right)\longmapsto \left(t_{1},\,\ldots ,\,t_{i-1},\,t_{i+1},\,,\ldots ,,\,t_{n}\right),$

as we can reconstruct

${}t_{i}=-{\frac {1}{f_{i}}}{\left(t_{1}f_{1}+\cdots +t_{i-1}f_{i-1}+t_{i+1}f_{i+1}+\cdots +t_{n}f_{n}\right)}\,$

from the other variables. This local trivialization also shows that, on the intersection ${}D(f_{i})\cap D(f_{j})$, the transformation between two trivializations is linear. So locally, the object ${}V{|}_{U}$ is trivial, but globally it might be complicated.

We now consider the scheme version of a vector bundle and in particular of a syzygy bundle (kernel bundle), trying to keep the idea that we are dealing with objects from linear algebra, but over a varying base. Let ${}R$ denote a commutative ring, let ${}I$ denote an ideal and fix generators ${}I={\left(f_{1},\ldots ,f_{n}\right)}$. This defines a short exact sequence of ${}R$-modules

$0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\longrightarrow R^{n}{\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}I\longrightarrow 0.$

The syzygy module ${}\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}$ is in general not locally free on ${}X=\operatorname {Spec} {\left(R\right)}$. However, on the open subset

${}U=D(I)=\bigcup _{i=1}^{n}D(f_{i})\subseteq X\,$

defined by the ideal, this module will be locally free, since it is free on each ${}D(f_{i})$, using the same trivialization as above. Moreover, on ${}U$ we get the short exact sequence

$0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}{|}_{U}\longrightarrow {\mathcal {O}}_{U}^{n}{\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{U}\longrightarrow 0$

of coherent sheaves. Later on, ${}I$ will be an ${}{\mathfrak {m}}$-primary ideal in a local noetherian ring ${}R$ and then

${}U=X\setminus \{{\mathfrak {m}}\}\,$

will be the punctured spectrum of ${}R$. The sheaves occurring in the last sequence are locally free in the following sense.

Definition

A coherent ${}{\mathcal {O}}_{X}$-module ${}{\mathcal {F}}$ on a scheme ${}X$ is called locally free of rank ${}r$, if there exists an open covering ${}X=\bigcup _{i\in I}U_{i}$ and ${}{\mathcal {O}}_{U_{i}}$-module-isomorphisms ${}{\mathcal {F}}{|}_{U_{i}}\cong {\left({\mathcal {O}}_{U_{i}}\right)}^{r}$ for every

${}i\in I$.

An equivalent concept of a locally free sheaf is the concept of a geometric vector bundle.

Definition

Let ${}X$ denote a scheme. A scheme ${}V$ equipped with a morphism

$p\colon V\longrightarrow X$

is called a geometric vector bundle of rank ${}r$ over ${}X$ if there exists an open covering ${}X=\bigcup _{i\in I}U_{i}$ and ${}U_{i}$-isomorphisms

$\psi _{i}\colon U_{i}\times {{\mathbb {A} }_{}^{r}}={{\mathbb {A} }_{U_{i}}^{r}}\longrightarrow V{|}_{U_{i}}=p^{-1}(U_{i})$

such that for every open affine subset ${}U\subseteq U_{i}\cap U_{j}$, the transition mappings

$\psi _{j}^{-1}\circ \psi _{i}\colon {{\mathbb {A} }_{U_{i}}^{r}}{|}_{U}\longrightarrow {{\mathbb {A} }_{U_{j}}^{r}}{|}_{U}$

are linear automorphisms, i.e. they are induced by an automorphism of the polynomial ring ${}\Gamma (U,{\mathcal {O}}_{X})[T_{1},\ldots ,T_{r}]$ given by ${}T_{i}\mapsto \sum _{j=1}^{r}a_{ij}T_{j}$.

We will work with both concepts and switch between them as needed.

The graded case

We will restrict now to the standard-graded case in order to work on the corresponding projective variety. Let ${}R$ be a standard-graded normal domain over an algebraically closed field ${}K$. Let

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

be the corresponding projective variety 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 ${}Y$ the short exact sequence

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

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

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

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

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

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

(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

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

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

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

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

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

for all ${}q$ and ${}m$, at least asymptotically. This is a Frobenius-Riemann-Roch problem.

By this translation of Hilbert-Kunz theory into a projective setting, we gain the following.

1. We work with projective varieties; if we look at local rings with an isolated singularity, we even work on smooth projective varieties.
2. We work with locally free sheaves. Taking Frobenius pull-backs is then exact. The mentioned Frobenius-Riemann-Roch problem is not specific for syzygy bundles, but should be addressed in general.
3. We can use the advanced methods of algebraic geometry, like intersection theory, Riemann-Roch theorem, vanishing theorems, ampleness, cohomology, moduli spaces.

This is still a difficult problem in general. However, if the local normal ring has dimension two and the corresponding variety is a smooth projective curve, then our understanding is good enough to solve the main problems from Hilbert-Kunz theory. The main advantages in the curve case compared to the case of higher-dimensional varieties are the following.

1. The degree of a vector bundle is independent of a polarization.
2. There are only the ${}0$th and the first cohomology, which are directly related by Serre-duality.
3. The Riemann-Roch theorem relates these notions.
4. Semistability gives good criteria for having no global sections.

We introduce these concepts on a smooth projective curve ${}C$ over an algebraically closed field ${}K$.

Definition

Let ${}C$ denote a smooth projective curve over an algebraically closed field ${}K$. For a locally free sheaf ${}{\mathcal {G}}$ on ${}C$ of rank ${}r$ we define its degree by the degree of the determinant sheaf

${}\bigwedge ^{r}{\mathcal {G}}$.

The determinant bundle is an invertible sheaf and corresponds therefore to a Weil divisor, say

${}D=\sum _{P\in C}n_{P}P\,,$

its degree is defined by ${}\sum _{P\in C}n_{P}$. The degree of the curve itself is defined as the degree of ${}{\mathcal {O}}_{C}(1)$. The degree of bundles is additive on short exact sequences of locally free sheaves. Applying additivity to

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

we get

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

Definition

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 ${}{\mathcal {T}}\subseteq {\mathcal {S}}$.

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. An important property of a semistable bundle of negative degree is that it can not have any global section ${}\neq 0$. The semistable bundles are those for which there exists a moduli space.

Example

Let ${}R=K[x,y,z]/{\left(x^{3}+y^{3}+z^{3}\right)}$, where ${}K$ is a field of positive characteristic ${}p\neq 3$, ${}I={\left(x^{2},y^{2},z^{2}\right)}$, and

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

The equation ${}x^{3}+y^{3}+z^{3}=0$ yields the short exact sequence

$0\longrightarrow {\mathcal {O}}_{C}\longrightarrow \operatorname {Syz} {\left(x^{2},y^{2},z^{2}\right)}(3)\longrightarrow {\mathcal {O}}_{C}\longrightarrow 0.$

This shows that ${}\operatorname {Syz} {\left(x^{2},y^{2},z^{2}\right)}$ is strongly semistable.

Example

Let ${}C$ be the smooth Fermat quartic given by ${}x^{4}+y^{4}+z^{4}$, and consider on it the syzygy bundle ${}\operatorname {Syz} {\left(x,y,z\right)}$ (which is also the restricted cotangent bundle from the projective plane). This bundle is semistable. Suppose that the characteristic is ${}3$. Then its Frobenius pull-back is ${}\operatorname {Syz} {\left(x^{3},y^{3},z^{3}\right)}$. The curve equation gives a global non-trivial section of this bundle of total degree ${}4$. But the degree of ${}\operatorname {Syz} {\left(x^{3},y^{3},z^{3}\right)}(4)$ is negative, hence it can not be semistable anymore.

Definition

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

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.

This theorem is due to A. Langer and holds also in higher dimension. 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 filtrations 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

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

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

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