Kurs:Vector bundles, forcing algebras and local cohomology (Tehran 2012)/Lecture 2

Geometric vector bundles

We have seen that the forcing algebra has locally the form ${}R_{f_{i}}[T_{1},\ldots ,T_{i-1},T_{i+1},\ldots ,T_{n}]$ and its spectrum ${}\operatorname {Spec} _{}^{}{\left(B\right)}$ has locally the form ${}D(f_{i})\times {\mathbb {A} }_{K}^{n-1}$. This description holds on the union ${}U=\bigcup _{i=1}^{n}D(f_{i})$. Moreover, in the homogeneous case ($f=0$) the transition mappings are linear. Hence ${}V{|}_{U}$ is a geometric vector bundle according to the following definition.

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 ${}T_{i}\mapsto \sum _{j=1}^{r}a_{ij}T_{j}$.

Here we can restrict always to affine open coverings. If ${}X$ is separated then the intersection of two affine open subschemes is again affine and then it is enough to check the condition on the intersections. The trivial bundle of rank ${}r$ is the ${}r$-dimensional affine space ${}{{\mathbb {A} }_{X}^{r}}$ over ${}X$, and locally every vector bundle looks like this. Many properties of an affine space are enjoyed by general vector bundles. For example, in the affine space we have the natural addition

$+\colon {{\mathbb {A} }_{U}^{r}}\times _{U}{{\mathbb {A} }_{U}^{r}}\longrightarrow {{\mathbb {A} }_{U}^{r}},\,(v_{1},\ldots ,v_{r},w_{1},\ldots ,w_{r})\longmapsto (v_{1}+w_{1},\ldots ,v_{r}+w_{r}),$

and this carries over to a vector bundle, that is, we have an addition

$\alpha \colon V\times _{X}V\longrightarrow V.$

The reason for this is that the isomorphisms occurring in the definition of a geometric vector bundle are linear, hence the addition on ${}V{|}_{U}$ coming from an isomorphism with some affine space over ${}U$ is independent of the choosen isomorphism. For the same reason there is a unique closed subscheme of ${}V$ called the zero-section which is locally defined to be ${}0\times U\subseteq {{\mathbb {A} }_{U}^{r}}$. Also, multiplication by a scalar, i.e. the mapping

$\cdot \colon {\mathbb {A} }_{U}^{1}\times _{U}{{\mathbb {A} }_{U}^{r}}\longrightarrow {{\mathbb {A} }_{U}^{r}},\,(s,v_{1},\ldots ,v_{r})\longmapsto (sv_{1},\ldots ,sv_{r}),$

carries over to a scalar multiplication

$\cdot \colon {\mathbb {A} }_{X}\times _{X}V\longrightarrow V.$

In particular, for every point ${}P\in X$ the fiber ${}V_{P}=V\times _{X}P$ is an affine space over ${}\kappa (P)$.

For a geometric vector bundle ${}p\colon V\rightarrow X$ and an open subset ${}U\subseteq X$ one sets

${}\Gamma (U,V)={\left\{s:U\rightarrow V{|}_{U}\mid p\circ s=\operatorname {Id} _{U}\right\}}\,,$

so this is the set of sections in ${}V$ over ${}U$. This gives in fact for every scheme over ${}X$ a set-valued sheaf. Because of the observations just mentioned, these sections can also be added and multiplied by elements in the structure sheaf, and so we get for every vector bundle a locally free sheaf, which is free on the open subsets where the vector bundle is trivial.

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 ${}i\in I$.

Vector bundles and locally free sheaves are essentially the same objects.

Theorem

Let ${}X$ denote a scheme.

Then the category of locally free sheaves on ${}X$ and the category of geometric vector bundles on ${}X$ are equivalent.

A geometric vector bundle ${}V\rightarrow X$ corresponds to the sheaf of its sections, and a locally free sheaf ${}{\mathcal {F}}$ corresponds to the (relative) spectrum of the symmetric algebra of the dual module ${}{\mathcal {F}}^{*}$.

The free sheaf of rank ${}r$ corresponds to the affine space ${}{{\mathbb {A} }_{X}^{r}}$ over ${}X$.

Torsors of vector bundles

We have seen that

${}V=\operatorname {Spec} {\left(R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}\right)}\right)}\,$

acts on the spectrum of a forcing algebra ${}T=\operatorname {Spec} {\left(R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}\right)}$ by addition. The restriction of ${}V$ to ${}U=D({\left(f_{1},\ldots ,f_{n}\right)}$ 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 is 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).

Forcing algebras and induced torsors

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.

In the next lectures we will deal with global properties of torsors and forcing algebras and how these properties are related to closure operations of ideals.

Exercise for Saturday: Show that ${}f$ belongs to the radical of the ideal ${}(f_{1},\ldots ,f_{n})$ if and only if the spectrum morphism

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

is surjective.

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