Kurs:Vector bundles and ideal closure operations (MSRI 2012)/Lecture 2

Torsors of vector bundles

We have seen that ${}V=\operatorname {Spec} _{}^{}{\left(R[T_{1},\ldots ,T_{n}]/(f_{1}T_{1}+\cdots +f_{n}T_{n})\right)}$ acts on the spectrum of a forcing algebra ${}T=\operatorname {Spec} _{}^{}{\left(R[T_{1},\ldots ,T_{n}]/(f_{1}T_{1}+\cdots +f_{n}T_{n}+f)\right)}$ by addition. The restriction of ${}V$ to ${}U=D(f_{1},\ldots ,f_{n})$ is a vector bundle, and ${}T$ restricted to ${}U$ becomes a ${}V$-torsor.

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 ${}V$-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.

Proof

We describe only the correspondence. Let ${}T$ denote a ${}V$-torsor. Then there exists by definition an open covering ${}X=\bigcup _{i\in I}U_{i}$ such that there exist isomorphisms

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

which are compatible with the action of ${}V{|}_{U_{i}}$ on itself. The isomorphisms ${}\varphi _{i}$ induce automorphisms

$\psi _{ij}=\varphi _{j}\circ \varphi _{i}^{-1}\colon V{|}_{U_{i}\cap U_{j}}\longrightarrow V{|}_{U_{i}\cap U_{j}}.$

These automorphisms are compatible with the action of ${}V$ on itself, and this means that they are of the form

${}\psi _{ij}=\operatorname {Id} _{V}{|}_{U_{i}\cap U_{j}}+s_{ij}\,$

with suitable sections ${}s_{ij}\in \Gamma (U_{i}\cap U_{j},{\mathcal {S}})$. This family defines a Čech cocycle for the covering and gives therefore a cohomology class in ${}H^{1}(X,{\mathcal {S}})$.
For the reverse direction, suppose that the cohomology class ${}c\in H^{1}(X,{\mathcal {S}})$ is represented by a Čech cocycle ${}s_{ij}\in \Gamma (U_{i}\cap U_{j},{\mathcal {S}})$ for an open covering ${}X=\bigcup _{i\in I}U_{i}$. Set ${}T_{i}:=V{|}_{U_{i}}$. We take the morphisms

$\psi _{ij}\colon T_{i}{|}_{U_{i}\cap U_{j}}=V{|}_{U_{i}\cap U_{j}}\longrightarrow V{|}_{U_{i}\cap U_{j}}=T_{j}{|}_{U_{i}\cap U_{j}}$

given by ${}\psi _{ij}:=\operatorname {Id} _{V}{|}_{U_{i}\cap U_{j}}+s_{ij}$ to glue the ${}T_{i}$ together to a scheme ${}T$ over ${}X$. This is possible since the cocycle condition guarantees the glueing condition for schemes.

The action of ${}T_{i}=V{|}_{U_{i}}$ on itself glues also together to give an action on ${}T$.

$\Box$

It follows immediately that for an affine scheme (i.e. a scheme of type $\operatorname {Spec} _{}^{}{\left(R\right)}$) there are no non-trivial torsor for any vector bundle. There will however be in general many non-trivial torsors on the punctured spectrum (and on a projective variety).

We mention an alternative description of the torsor corresponding to a first cohomology class in a locally free sheaf which is better suited for the projective situation.

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.

Forcing algebras and induced torsors

Let ${}T=\operatorname {Spec} _{}^{}{\left(B\right)}$ be the spectrum of a forcing algebra. As ${}T{|}_{U}$ is a ${}V{|}_{U}$-torsor, and as every ${}V$-torsor is represented by a unique cohomology class, there should be a natural cohomology class coming from the forcing data. To see this, let ${}R$ be a noetherian ring and ${}I=(f_{1},\ldots ,f_{n})$ be an ideal. Then on ${}U=D(I)$ we have the short exact sequence

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

An element ${}f\in R$ defines an element ${}f\in \Gamma (U,{\mathcal {O}}_{U})$ and hence a cohomology class ${}\delta (f)\in H^{1}(U,\operatorname {Syz} _{}^{}{\left(f_{1},\ldots ,f_{n}\right)})$. Hence ${}f$ defines in fact a ${}\operatorname {Syz} _{}^{}{\left(f_{1},\ldots ,f_{n}\right)}$-torsor over ${}U$. We will see that this torsor is induced by the forcing algebra given by ${}f_{1},\ldots ,f_{n}$ and ${}f$.

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 ${}c\in H^{1}(D(I),\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})$ 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 ${}c$.

Proof

We compute the cohomology class ${}\delta (f)\in H^{1}(U,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})$ and the cohomology class given by the forcing algebra. For the first computation we look at the short exact sequence

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

On ${}D(f_{i})$, the element ${}f$ is the image of ${}\left(0,\ldots ,0,\,{\frac {f}{f_{i}}},\,0,\ldots ,0)\right)$ (the non-zero entry is at the ${}i$th place). The cohomology class is therefore represented by the family of differences

$\left(0,\ldots ,0,\,{\frac {f}{f_{i}}},\,0,\ldots ,0,\,-{\frac {f}{f_{j}}},\,0,\ldots ,0\right)\in \Gamma (D(f_{i})\cap D(f_{j}),\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}).$

On the other hand, there are isomorphisms

$V{|}_{D(f_{i})}\longrightarrow T{|}_{D(f_{i})},\,\left(s_{1},\ldots ,s_{n}\right)\longmapsto \left(s_{1},\ldots ,s_{i-1},\,s_{i}+{\frac {f}{f_{i}}},\,s_{i+1},\ldots ,s_{n}\right).$

The composition of two such isomorphisms on ${}D(f_{i}f_{j})$ is the identity plus the same section as before.

$\Box$

Example

Let ${}(R,{\mathfrak {m}})$ denote a two-dimensional normal local noetherian domain and let ${}f$ and ${}g$ be two parameters in ${}R$. On ${}U=D({\mathfrak {m}})$ we have the short exact sequence

$0\longrightarrow {\mathcal {O}}_{U}\cong \operatorname {Syz} {\left(f,g\right)}\longrightarrow {\mathcal {O}}_{U}^{2}{\stackrel {f,g}{\longrightarrow }}{\mathcal {O}}_{U}\longrightarrow 0$

and its corresponding long exact sequence of cohomology,

$0\longrightarrow R\longrightarrow R^{2}{\stackrel {f,g}{\longrightarrow }}R{\stackrel {\delta }{\longrightarrow }}H^{1}(U,{\mathcal {O}}_{X})\longrightarrow \ldots .$

The connecting homomorphism ${}\delta$ sends an element ${}h\in R$ to ${}{\frac {h}{fg}}$. The torsor given by such a cohomology class ${}c={\frac {h}{fg}}\in H^{1}{\left(U,{\mathcal {O}}_{X}\right)}$ can be realized by the forcing algebra

$R[T_{1},T_{2}]/{\left(fT_{1}+gT_{2}-h\right)}.$

Note that different forcing algebras may give the same torsor, because the torsor depends only on the spectrum of the forcing algebra restricted to the punctured spectrum of ${}R$. For example, the cohomology class ${}{\frac {1}{fg}}={\frac {fg}{f^{2}g^{2}}}$ defines one torsor, but the two fractions yield the two forcing algebras ${}R[T_{1},T_{2}]/{\left(fT_{1}+gT_{2}-1\right)}$ and ${}R[T_{1},T_{2}]/{\left(f^{2}T_{1}+g^{2}T_{2}-fg\right)}$, which are quite different. The fiber over the maximal ideal of the first one is empty, whereas the fiber over the maximal ideal of the second one is a plane.

If ${}R$ is regular, say ${}R=K[X,Y]$ (or the localization of this at $(X,Y)$ or the corresponding power series ring) then the first cohomology classes are ${}K$-linear combinations of ${}{\frac {1}{x^{i}y^{j}}}$, ${}i,j\geq 1$.

They are realized by the forcing algebras

$K[X,Y,T_{1},T_{2}]/{\left(X^{i}T_{1}+Y^{j}T_{2}-1\right)}.$

Since the fiber over the maximal ideal is empty, the spectrum of the forcing algebra equals the torsor. Or, the other way round, the torsor is itself an affine scheme.

The closure operations we have considered in the first lecture and in the tutorial can be characterized by some property of the forcing algebra. However, they can not be characterized by a property of the corresponding torsor alone. For example, for ${}R=K[X,Y]$, we may write

${}{\frac {1}{XY}}={\frac {X}{X^{2}Y}}={\frac {XY}{X^{2}Y^{2}}}={\frac {X^{2}Y^{2}}{X^{3}Y^{3}}}\,,$

so the torsors given by the forcing algebras

$R[T_{1},T_{2}]/(XT_{1}+YT_{2}+1),\,R[T_{1},T_{2}]/(X^{2}T_{1}+YT_{2}+X),\,R[T_{1},T_{2}]/(X^{2}T_{1}+Y^{2}T_{2}+XY){\text{ and }}R[T_{1},T_{2}]/(X^{3}T_{1}+Y^{3}T_{2}+X^{2}Y^{2})$

are all the same (the restriction over $D(X,Y)$), but their global properties are quite different. We have a non-surjection, a surjective non submersion, a submersion which does not admit (for $K={\mathbb {C} }$) a continuous section and a map which admits a continuous section.

Tight closure and solid closure

We deal now with closure operations which depend only on the torsor which the forcing algebra defines, so they only depend on the cohomology class of the forcing data inside the syzygy bundle. Our main example is tight closure, a theory developed by Hochster and Huneke, and related closure operations like solid closure and plus closure.

Let ${}R$ be a noetherian domain of positive characteristic, let

$F\colon R\longrightarrow R,\,f\longmapsto f^{p},$

be the Frobenius homomorphism and

$F^{e}\colon R\longrightarrow R,\,f\longmapsto f^{q}$

(mit ${}q=p^{e}$) its ${}e$th iteration. Let ${}I$ be an ideal and set

${}I^{[q]}={\text{ extended ideal of }}I{\text{ under }}F^{e}\,.$

Then define the tight closure of ${}I$ to be the ideal

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

The element ${}f$ defines the cohomology class ${}c\in H^{1}(D(I),\operatorname {Syz} _{}^{}{\left(f_{1},\ldots ,f_{n}\right)})$. Suppose that ${}R$ is normal and that ${}I$ has height at least ${}2$ (think of a local normal domain of dimension at least ${}2$ and an ${}{\mathfrak {m}}$-primary ideal ${}I$). Then the ${}e$th Frobenius pull-back of the cohomology class is

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

($q=p^{e}$) and this is the cohomology class corresponding to ${}f^{q}$. By the height assumption, ${}zF^{e*}(c)=0$ if and only if ${}zf^{q}\in (f_{1}^{q},\ldots ,f_{n}^{q})$, and if this holds for all ${}e$ then ${}f\in I^{*}$ by definition. This shows already that tight closure under the given conditions does only depend on the cohomology class.

This is also a consequence of the following theorem of Hochster which 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

${}H_{\mathfrak {m}}^{\dim(R)}(B)\neq 0\,,$

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

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 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 this equivalence - 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.

In the following we will deal with some important questions from tight closure theory, namely the relation to plus closure and whether tight closure commutes with localization. The geometric interpretation with bundles will guide us through these questions.

Plus closure

For an ideal ${}I\subseteq R$ in a domain ${}R$ define its plus closure by

${}I^{+}={\left\{f\in R\mid {\text{there exists a finite domain extension }}R\subseteq T{\text{ such that }}f\in IT\right\}}\,.$

Equivalent: Let ${}R^{+}$ be the absolute integral closure of ${}R$. This is the integral closure of ${}R$ in an algebraic closure of the quotient field ${}Q(R)$ (first considered by Artin). Then

$f\in I^{+}{\text{ if and only if }}f\in IR^{+}.$

The plus closure commutes with localization.

We also have the inclusion ${}I^{+}\subseteq I^{*}$. Here the question arises:

Question: Is ${}I^{+}=I^{*}$?

This question is known as the tantalizing question in tight closure theory.

In terms of forcing algebras and their torsors, the containment inside the plus closure means that there exists a ${}d$-dimensional closed subscheme inside the spectrum of the forcing algebra which meets the exceptional fiber (the fiber over the maximal ideal) in only finitely many points. This also means that the superheight of the extended ideal is ${}d$. In this case the local cohomological dimension of the torsor must be ${}d$ as well, since it contains a closed subscheme with this cohomological dimension. So also the plus closure depends only on the torsor. One should think of tight closure as a cohomological property and of plus closure as a geometric property of torsors.

One of the main results about tight closure and plus closure is the following theorem due to K. Smith.

Theorem

Let ${}R$ be local and excellent. If ${}I$ is a parameter ideal (generated by a (sub-)system of parameters), then

$I^{*}=I^{+}\,$

and the tight closure of $I$ commutes with localization.

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