Vector bundles and tight closure (Triest 2023)/Lecture 2

Tight closure

Definition

Let ${}R$ be a noetherian domain of positive characteristic and let ${}I\subseteq R$ be an ideal. The tight closure of ${}I$ is the ideal

${}I^{*}={\left\{f\in R\mid {\text{ there exists }}z\neq 0{\text{ such that }}zf^{q}\in I^{[q]}{\text{ for all }}q=p^{e}\right\}}\,.$

This theory was introduced by M. Hochster and C. Huneke. There is a direct relation between Hilbert-Kunz multiplicity and tight closure.

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 try to understand tight closure from the perspective of bundles and will have again a look at the syzygy bundle. Let ${}R$ denote a noetherian normal domain and let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ denote an ideal of height ${}I$ at least ${}2$ (think of a local normal domain of dimension at least ${}2$ and an ${}{\mathfrak {m}}$-primary ideal ${}I$, or the graded version of this). Let ${}U=D(I)\subseteq \operatorname {Spec} {\left(R\right)}$ and consider again 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 locally free sheaves on ${}U$. Another element ${}f\in R=\Gamma (U,{\mathcal {O}}_{U})$ (because of the height condition) defines via the long exact sequence of cohomology the cohomology class ${}c=\delta (f)\in H^{1}(U,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})$. When ${}R$ contains a field of positive characteristic, we try to understand tight closure in terms of this cohomology class. Quite directly, we have the ${}e$th absolute Frobenius on ${}U$. As the sheaves are locally free, we have

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

and the ${}e$th Frobenius pull-back of the cohomology class is

${}F^{e*}(c)\in H^{1}(D(I),F^{e*}{\left(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\right)}\cong H^{1}(D(I),\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)})\,$

(${}q=p^{e}$), and this is the cohomology class corresponding to ${}f^{q}$. By the height assumption, we have ${}zF^{e*}(c)=0$ if and only if ${}zf^{q}\in {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}$, and this holds for all ${}e$ if and only if ${}f\in I^{*}$ by definition. This shows already that under the given conditions, tight closure does only depend on the cohomology class. In the graded case, we can also translate the tight closure question ${}f\in I^{*}$ for homogeneous data into the question whether the corresponding cohomology class ${}c\in H^{1}(Y,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m))$ on ${}\operatorname {Proj} {\left(R\right)}$ is tightly zero in the sense that ${}zF^{e*}(c)=0$ holds for some homogenous ${}z\neq 0$ (${}z$ considered inside some ample invertible sheaf ${}{\mathcal {O}}_{Y}(\ell )$). This property of being tightly zero is relevant for every cohomology class in any locally free sheaf. Here, this translation is in particular helpful for inclusion results. For exclusion results we have to go another way and consider torsors.

Torsors and forcing algebras

We come back to the situation of a system of linear homogeneous equations over a field ${}K$ with which we tried to motivate the concept of a vector bundle. However, we now consider a system of linear inhomogeneous equations,

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

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

$\vdots$

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

The solution set ${}T$ of this inhomogeneous system may be empty, but nevertheless it is tightly related to the solution space of the homogeneous system. First of all, there exists an action

$V\times T\longrightarrow T,(v,t)\longmapsto v+t,$

because the sum of a solution of the homogeneous system and of a solution of the inhomogeneous system is again a solution of the inhomogeneous system. This action is a group action of the group ${}(V,+,0)$ on the set ${}T$. Moreover, if we fix one solution ${}t_{0}\in T$ (supposing that at least one solution exists), then there exists a bijection

$V\longrightarrow T,v\longmapsto v+t_{0}.$

This means that the group ${}V$ acts simply transitive on ${}T$, and so ${}T$ can be identified with the vector space ${}V$, however not in a canonical way.

Suppose now that ${}X$ is a geometric object and we have functions

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

on ${}X$ (which are continuous, or differentiable, or algebraic). As before, we get for the ${}f_{ij}$ a bundle with an addition and such that the fibers are vector spaces.

Then we can form the set

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

with the projection to ${}X$. Again, every fiber ${}T_{P}$ of ${}T$ over a point ${}P\in X$ is the solution set to the system of inhomogeneous linear equations which arises by inserting ${}P$ into ${}f_{ij}$ and ${}f_{i}$. The actions of the fibers ${}V_{P}$ on ${}T_{P}$ (coming from linear algebra) extend to an action

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

Also, if a (continuous, differentiable, algebraic) map

$s\colon X\longrightarrow T$

with ${}s(P)\in T_{P}$ exists, then we can construct a (continuous, differentiable, algebraic) isomorphism between ${}V$ and ${}T$. However, different from the situation in linear algebra (which corresponds to the situation where ${}X$ is just one point), such a section does rarely exist.

These objects ${}T$ have new and sometimes difficult global properties which we try to understand. We will work mainly in an algebraic setting and restrict to the situation where just one equation

${}f_{1}T_{1}+\cdots +f_{n}T_{n}=f\,$

is given. Then in the homogeneous case (${}f=0$) the fibers are vector spaces of dimension ${}n-1$ or ${}n$, and the later holds exactly for the points ${}P\in X$ where ${}f_{1}(P)=\ldots =f_{n}(P)=0$. In the inhomogeneous case the fibers are either empty or of dimension ${}n-1$ or ${}n$. We give a typical example.

Example

Let ${}X$ denote a plane (like $K^{2},\mathbb {R} ^{2},{\mathbb {A} }_{K}^{2}$) with coordinate functions ${}x$ and ${}y$. We consider an inhomogeneous linear equation of type

${}x^{a}t_{1}+y^{b}t_{2}=x^{c}y^{d}\,.$

The fiber of the solution set ${}T$ over a point ${}\neq (0,0)$ is one-dimensional, whereas the fiber over ${}(0,0)$ has dimension two (for ${}a,b,c,d\geq 1$). Many properties of ${}T$ depend on these four exponents.

In (most of) these example, we can observe the following behavior. On an open subset, the dimension of the fibers is constant and equals ${}n-1$, whereas the fiber over some special points degenerates to an ${}n$-dimensional solution set (or becomes empty).

Definition

Let ${}V$ denote a geometric vector bundle over a scheme ${}X$. A scheme ${}T\rightarrow X$ together with an action

$\beta \colon V\times _{X}T\longrightarrow T$

is called a geometric (Zariski)-torsor for ${}V$ (or a principal fiber bundle or a principal homogeneous space) if there exists an open covering ${}X=\bigcup _{i\in I}U_{i}$ and isomorphisms

$\varphi _{i}\colon T{|}_{U_{i}}\longrightarrow V{|}_{U_{i}}$

such that the diagrams (we set ${}U=U_{i}$ and ${}\varphi =\varphi _{i}$)

${\begin{matrix}V{|}_{U}\times _{U}T{|}_{U}&{\stackrel {\beta }{\longrightarrow }}&T{|}_{U}&\\\!\!\!\!\!\operatorname {Id} \times \varphi \downarrow &&\downarrow \varphi \!\!\!\!\!&\\V{|}_{U}\times _{U}V{|}_{U}&{\stackrel {\alpha }{\longrightarrow }}&V{|}_{U}&\!\!\!\!\!\\\end{matrix}}$

commute, where ${}\alpha$ is the addition on the vector bundle.

The torsors of vector bundles can be classified in the following way.

Proposition

Let ${}X$ denote a noetherian separated scheme and let

$p\colon V\longrightarrow X$

denote a geometric vector bundle on ${}X$ with sheaf of sections ${}{\mathcal {S}}$. Then there exists a correspondence between first cohomology classes ${}c\in H^{1}(X,{\mathcal {S}})$ and geometric ${}V$-torsors.

Remark

Let ${}{\mathcal {S}}$ denote a locally free sheaf on a scheme ${}X$. For a cohomology class ${}c\in H^{1}(X,{\mathcal {S}})$ one can construct a geometric object: Because of ${}H^{1}(X,{\mathcal {S}})\cong \operatorname {Ext} ^{1}({\mathcal {O}}_{X},{\mathcal {S}})$, the class defines an extension

$0\longrightarrow {\mathcal {S}}\longrightarrow {{\mathcal {S}}'}\longrightarrow {\mathcal {O}}_{X}\longrightarrow 0.$

This extension is such that under the connecting homomorphism of cohomology, ${}1\in \Gamma (X,{\mathcal {O}}_{X})$ is sent to ${}c\in H^{1}(X,{\mathcal {S}})$. The extension yields a projective subbundle

${}{\mathbb {P} }({\mathcal {S}}^{\vee })\subset {\mathbb {P} }({{\mathcal {S}}'}^{\vee })\,.$

If ${}V$ is the corresponding geometric vector bundle of ${}{\mathcal {S}}$, one may think of ${}{\mathbb {P} }({\mathcal {S}}^{\vee })$ as ${}{\mathbb {P} }(V)$ which consists for every base point ${}x\in X$ of all the lines in the fiber ${}V_{x}$ passing through the origin. The projective subbundle ${}{\mathbb {P} }(V)$ has codimension one inside ${}{\mathbb {P} }(V')$, for every point it is a projective space lying (linearly) inside a projective space of one dimension higher. The complement is then over every point an affine space. One can show that the global complement

${}T={\mathbb {P} }({{\mathcal {S}}'}^{\vee })\setminus {\mathbb {P} }({\mathcal {S}}^{\vee })\,$

is another model for the torsor given by the cohomology class. The advantage of this viewpoint is that we may work, in particular when ${}X$ is projective, in an entirely projective setting.

Within the algebraic setting, torsors can also be realized as open subsets of spectra of forcing algebras.

Definition

Let ${}R$ be a commutative ring and let ${}f_{1},\ldots ,f_{n}$ and ${}f$ be elements in ${}R$. Then the ${}R$-algebra

$R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}-f\right)}$

is called the forcing algebra of these elements

(or these data).

Theorem

Let ${}R$ denote a noetherian ring, let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ denote an ideal and let ${}f\in R$ be another element. Let ${}c=\delta (f)\in H^{1}(D(I),\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})$ be the corresponding cohomology class and let

${}B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}-f\right)}\,$

denote the forcing algebra for these data. Then the scheme ${}\operatorname {Spec} {\left(B\right)}{|}_{D(I)}$ together with the natural action of the syzygy bundle on it is isomorphic to the torsor given by ${}c$.

Forcing algebras provide a natural framework for closure operations in general, it is however a special feature of tight closure that the induced torsor contains the relevant information.

Tight closure and solid closure

Forcing algebras occurred in the work of Hochster on solid closure. The following theorem of Hochster gives a characterization of tight closure in terms of forcing algebra and local cohomology.

Theorem

Let ${}R$ be a normal excellent local domain with maximal ideal ${}{\mathfrak {m}}$ over a field of positive characteristic. Let ${}f_{1},\ldots ,f_{n}$ generate an ${}{\mathfrak {m}}$-primary ideal ${}I$ and let ${}f$ be another element in ${}R$. Then ${}f\in I^{*}$ if and only if ${}H_{\mathfrak {m}}^{\dim(R)}(A)\neq 0$, where ${}A=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}$ denotes the forcing algebra of these elements.

If the dimension ${}d$ is at least two, then

$H_{\mathfrak {m}}^{d}(R)\longrightarrow H_{\mathfrak {m}}^{d}(B)\cong H_{{\mathfrak {m}}B}^{d}(B)\cong H^{d-1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B}).$

This means that we have to look at the cohomological properties of the complement of the exceptional fiber over the closed point, i.e. the torsor given by these data. If ${}H^{d-1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0$ then this is true for all quasicoherent sheaves instead of just the structure sheaf. This property can be expressed by saying that the cohomological dimension of ${}D({\mathfrak {m}}B)$ is ${}\leq d-2$ and thus smaller than the cohomological dimension of the punctured spectrum ${}D({\mathfrak {m}})$, which is exactly ${}d-1$. So belonging to tight closure can be rephrased by saying that the formation of the corresponding torsor does not change the cohomological dimension.

If the dimension is two, then we have to look whether the first cohomology of the structure sheaf vanishes. This is true (by Serre's cohomological criterion for affineness) if and only if the open subset ${}D({\mathfrak {m}}B)$ is an affine scheme (the spectrum of a ring).

The right hand side of the equivalence in Theorem 2.9 (the non-vanishing of the top-dimensional local cohomology) is independent of any characteristic assumption, and can be taken as the basis for the definition of another closure operation, called solid closure. So the theorem above says that in positive characteristic, tight closure and solid closure coincide. There is also a definition of tight closure for algebras over a field of characteristic ${}0$ by reduction to positive characteristic.

The graded two-dimensional case

In the situation of a forcing algebra of homogeneous elements, this torsor ${}T$ can also be obtained as ${}\operatorname {Proj} {\left(B\right)}$, where ${}B$ is the (not necessarily positively) graded forcing algebra. In particular, it follows that the containment ${}f\in I^{*}$ is equivalent to the property that ${}T$ is not an affine variety. For this properties, positivity (ampleness) properties of the syzygy bundle are crucial. We need again the concept of semistability introduced in the first lecture.

For a strongly semistable vector bundle ${}{\mathcal {S}}$ on ${}C$ and a cohomology class ${}c\in H^{1}(C,{\mathcal {S}})$ with corresponding torsor we obtain the following affineness criterion.

Theorem

Let ${}C$ denote a smooth projective curve over an algebraically closed field ${}K$ and let ${}{\mathcal {S}}$ be a strongly semistable vector bundle over ${}C$ together with a cohomology class ${}c\in H^{1}(C,{\mathcal {S}})$. Then the torsor ${}T(c)$ is an affine scheme if and only if ${}\operatorname {deg} {\left({\mathcal {S}}\right)}<0$ and ${}c\neq 0$ (${}F^{e}(c)\neq 0$ for all ${}e$ in positive characteristic).

This result rests on the ampleness of ${}{\mathcal {S}}'^{\vee }$ occuring in the dual exact sequence ${}0\rightarrow {\mathcal {O}}_{C}\rightarrow {\mathcal {S}}'^{\vee }\rightarrow {\mathcal {S}}^{\vee }\rightarrow 0$ given by ${}c$ (this rests on work of Gieseker and Hartshorne). It implies for a strongly semistable syzygy bundle the following degree formula for tight closure.

Theorem

Suppose that ${}\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}$ is strongly semistable. Then

$R_{m}\subseteq I^{*}{\text{ for }}m\geq {\frac {\sum d_{i}}{n-1}}{\text{ and (for almost all prime numbers) }}R_{m}\cap I^{*}\subseteq I{\text{ for }}m<{\frac {\sum d_{i}}{n-1}}.$

If we take on the right hand side ${}I^{F}$, the Frobenius closure of the ideal, instead of ${}I$, then this statement is true for all characteristics. As stated, it is true in a relative setting for ${}p$ large enough.

We indicate the proof of the inclusion result. The degree condition implies that ${}c\in \delta (f)=H^{1}(C,{\mathcal {S}})$ is such that ${}{\mathcal {S}}=\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)$ has non-negative degree. Then also all Frobenius pull-backs ${}F^{*}({\mathcal {S}})$ have non-negative degree. Let ${}{\mathcal {L}}={\mathcal {O}}(k)$ be a twist of the tautological line bundle on ${}C$ such that its degree is larger than the degree of ${}\omega _{C}^{-1}$, the dual of the canonical sheaf. Let ${}z\in H^{0}(Y,{\mathcal {L}})$ be a non-zero element. Then ${}zF^{e*}(c)\in H^{1}(C,F^{e*}({\mathcal {S}})\otimes {\mathcal {L}})$, and by Serre duality we have

${}H^{1}(C,F^{e*}({\mathcal {S}})\otimes {\mathcal {L}})\cong H^{0}(F^{e*}({\mathcal {S}}^{\vee })\otimes {\mathcal {L}}^{-1}\otimes \omega _{C})^{\vee }\,.$

On the right hand side we have a semistable sheaf of negative degree, which can not have a non-trivial section. Hence

${}zF^{e*}(c)=0\,,$

and therefore ${}f$ belongs to the tight closure.

In general, there exists an exact criterion for the affineness of the torsor ${}T(c)$ depending on ${}c$ and the strong Harder-Narasimhan filtration of ${}{\mathcal {S}}$.

Theorem

Let ${}C$ denote a smooth projective curve over an algebraically closed field ${}K$ and let ${}{\mathcal {S}}$ be a vector bundle over ${}C$ together with a cohomology class ${}c\in H^{1}(C,{\mathcal {S}})$. Let

${}{\mathcal {S}}_{1}\subset {\mathcal {S}}_{2}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}=F^{e*}({\mathcal {S}})\,$

be a strong Harder-Narasimhan filtration. We choose ${}i$ such that ${}{\mathcal {S}}_{i}/{\mathcal {S}}_{i-1}$ has degree ${}\geq 0$ and that ${}{\mathcal {S}}_{i+1}/{\mathcal {S}}_{i}$ has degree ${}<0$. We set

${}{\mathcal {Q}}=F^{e*}({\mathcal {S}})/{\mathcal {S}}_{i}$. Then the following are equivalent.

1. The torsor ${}T(c)$ is not an affine scheme.
2. Some Frobenius power of the image of ${}F^{e*}(c)$ inside ${}H^{1}(X,{\mathcal {Q}})$ is ${}0$.