Kurs:Vektorbündel und Abschlussoperationen (MSRI 2012)

Vector bundles and ideal closure operations

The following notes give an introduction to the use of vector bundle techniques to ideal closure operations, in particular to tight closure and related closures like solid closure and plus closure. They are also an introduction to the theory of vector bundles in general with an emphasis on smooth projective curves. Moreover, they give an introduction to the relationship between forcing algebras and closure operations. Throughout we assume a basic knowledge of commutative algebra and algebraic geometry including local cohomology and sheaf cohomology; once in a while we will use some notions and results from tight closure theory.

These notes are based on lectures which I gave on several occasions in Barcelona, Kolkatta, Tehran, Medellín, Ann Arbor and at MSRI during the Introductory Workshop for the Special Year on Commutative Algebra 2012-2013. I thank the organizers of the Special Year, David Eisenbud, Srikanth Iyengar, Ezra Miller, Anurag Singh and Karen Smith for the opportunity to be part in the Introductory Workshop. I thank MSRI for its hospitality during the workshop and in the spring semester 2013. I thank Matteo Varbaro who gave the tutorials for the lectures at MSRI.

For corrections and remarks I thank Mohsen Asgharzadeh, Daniel Brinkmann, Amartya Dutta, Neena Gupta, Axel Stäbler, Matteo Varbaro and Juan Vélez.

These lectures were prepared and written on the German Wikiversity and are under the CC-by-sa 3.0 licence. This basically means that everybody is allowed to copy it, use it and make it better (also in parts).

Linear equations, forcing algebras and closure operations

Systems of linear equations

We start with some 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 subvector space of $K^{n}$), its dimension is ${}n-\operatorname {rk} (A)$, where

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

is the matrix given by these elements. Additional elements ${}f_{1},\ldots ,f_{m}\in K$ give rise to the system of inhomogeneous linear 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 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 (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 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

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

(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.

Suppose now that additional functions

$f_{1},\ldots ,f_{m}\colon X\longrightarrow K$

are given. 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 in these lectures. 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 some typical examples.

Example

We consider the line ${}X={\mathbb {A} }_{K}^{1}$ (or ${}X=K,\mathbb {R} ,{\mathbb {C} }$ etc.) with the (identical) function ${}x$. For ${}f_{1}=x$ and ${}f=0$, i.e. for the homogeneous equation ${}xt=0$, the geometric object ${}V$ consists of a horizontal line (corresponding to the zero-solution) and a vertical line over ${}x=0$. So all fibers except one are zero-dimensional vector spaces. For the inhomogeneous equation ${}xt=1$, ${}T$ is a hyperbola, and all fibers are zero-dimensional with the exception that the fiber over ${}x=0$ is empty.

For the homogeneous equation ${}0t=0$, ${}V$ is just the affine cylinder over the base line. For the inhomogeneous equation ${}0t=x$, ${}T$ consists of one vertical line, almost all fibers are empty.

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).

Forcing algebras

We describe now the algebraic setting of systems of linear equations depending on a base space. For a commutative ring ${}R$, its spectrum ${}X=\operatorname {Spec} {\left(R\right)}$ is a topological space on which the ring elements can be considered as functions. The value of ${}f\in R$ at a prime ideal ${}P\in \operatorname {Spec} {\left(R\right)}$ is just the image of ${}f$ under the ring homomorphism ${}R\rightarrow R/P\rightarrow \kappa (P)=Q(R/P)$. In this interpretation, a ring element is a function with values in different fields. Suppose that ${}R$ contains a field ${}K$. Then an element ${}f\in R$ gives rise to the ring homomorphism

$K[Y]\longrightarrow R,\,Y\longmapsto f,$

which gives rise to a scheme morphism

$\operatorname {Spec} {\left(R\right)}\longrightarrow \operatorname {Spec} {\left(K[Y]\right)}\cong {\mathbb {A} }_{K}^{1}.$

This is another way to consider ${}f$ as a function on ${}\operatorname {Spec} {\left(R\right)}$ with values in the affine line.

The following construction appeared first in the work of Hochster in the context of solid closure.

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).

The forcing algebra ${}B$ forces ${}f$ to lie inside the extended ideal ${}{\left(f_{1},\ldots ,f_{n}\right)}B$ (hence the name). For every ${}R$-algebra ${}S$ such that ${}f\in {\left(f_{1},\ldots ,f_{n}\right)}S$ there exists a (non unique) ring homomorphism ${}B\rightarrow S$ by sending ${}T_{i}$ to the coefficient ${}s_{i}\in S$ in an expression ${}f=s_{1}f_{1}+\cdots +s_{n}f_{n}$.

The forcing algebra induces the spectrum morphism

$\operatorname {Spec} {\left(B\right)}\longrightarrow \operatorname {Spec} {\left(R\right)}.$

Over a point ${}P\in X=\operatorname {Spec} {\left(R\right)}$, the fiber of this morphism is given by

$\operatorname {Spec} {\left(B\otimes _{R}\kappa (P)\right)},$

and we can write

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

where ${}f_{i}(P)$ means the evaluation of the ${}f_{i}$ in the residue class field. Hence the ${}\kappa (P)$-points in the fiber are exactly the solutions to the inhomogeneous linear equation ${}f_{1}(P)T_{1}+\cdots +f_{n}(P)T_{n}=f(P)$. In particular, all the fibers are (empty or) affine spaces.

Forcing algebras and closure operations

Let ${}R$ denote a commutative ring and let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ideal. Let ${}f\in R$ and let

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

be the corresponding forcing algebra and

$\varphi \colon \operatorname {Spec} {\left(B\right)}\longrightarrow \operatorname {Spec} {\left(R\right)}$

the corresponding spectrum morphism. How are properties of ${}\varphi$ (or of the ${}R$-algebra ${}B$) related to certain ideal closure operations?

We start with some examples. The element ${}f$ belongs to the ideal ${}I$ if and only if we can write ${}f=r_{1}f_{1}+\cdots +r_{n}f_{n}$ with ${}r_{i}\in R$. By the universal property of the forcing algebra this means that there exists an ${}R$-algebra-homomorphism

$B\longrightarrow R,$

hence ${}f\in I$ holds if and only if ${}\varphi$ admits a scheme section. This is also equivalent to

$R\longrightarrow B$

admitting an ${}R$-module section or ${}B$ being a pure ${}R$-algebra (so for forcing algebras properties might be equivalent which are not equivalent for arbitrary algebras).

The radical of an ideal

Now we look at the radical of the ideal ${}I$,

${}\operatorname {rad} {\left(I\right)}={\left\{f\in R\mid f^{k}\in I{\text{ for some }}k\right\}}\,.$

The importance of the radical comes mainly from Hilbert's Nullstellensatz, saying that for algebras of finite type over an algebraically closed field there is a natural bijection between radical ideals and closed algebraic zero-sets. So geometrically one can see from an ideal only its radical. As this is quite a coarse closure operation we should expect that this corresponds to a quite coarse property of the morphism ${}\varphi$ as well. Indeed, it is true that ${}f\in \operatorname {rad} {\left(I\right)}$ if and only if ${}\varphi$ is surjective. This is true since the radical of an ideal is the intersection of all prime ideals in which it is contained. Hence an element ${}f$ belongs to the radical if and only if for all residue class homomorphisms

$\theta \colon R\longrightarrow \kappa ({\mathfrak {p}})$

where ${}I$ is sent to ${}0$, also ${}f$ is sent to ${}0$. But this means for the forcing equation that whenever the equation degenerates to ${}0$, then also the inhomogeneous part becomes zero, and so there will always be a solution to the inhomogeneous equation.

Exercise: Define the radical of a submodule inside a module.

Integral closure of an ideal

Another closure operation is integral closure. It is defined by

${}{\overline {I}}={\left\{f\in R\mid f^{k}+a_{1}f^{k-1}+\cdots +a_{k-1}f+a_{k}=0{\text{ for some }}k{\text{ and }}a_{i}\in I^{i}\right\}}\,.$

This notion is important for describing the normalization of the blow up of the ideal ${}I$. Another characterization (assume that ${}R$ is noetherian) is that there exists a ${}z\in R$, not contained in any minimal prime ideal of ${}R$, such that ${}zf^{n}\in I^{n}$ holds for all ${}n$. Another equivalent property - the valuative criterion - is that for all ring homomorphisms

$\theta \colon R\longrightarrow D$

to a discrete valuation domain ${}D$ the containment ${}\theta (f)\in \theta (I)D$ holds.

The characterization of the integral closure in terms of forcing algebras requires some notions from topology. A continuous map

$\varphi \colon X\longrightarrow Y$

between topological spaces ${}X$ and ${}Y$ is called a submersion, if it is surjective and if ${}Y$ carries the image topology (quotient topology) under this map. This means that a subset ${}W\subseteq Y$ is open if and only if its preimage ${}\varphi ^{-1}(W)$ is open. Since the spectrum of a ring endowed with the Zarisiki topology is a topological space, this notion can be applied to the spectrum morphism of a ring homomorphism. With this notion we can state that ${}f\in {\bar {I}}$ if and only if the forcing morphism

$\varphi \colon \operatorname {Spec} {\left(B\right)}\longrightarrow \operatorname {Spec} {\left(R\right)}$

is a universal submersion (universal means here that for any ring change ${}R\rightarrow R'$ to a noetherian ring ${}R'$, the resulting homomorphism ${}R'\rightarrow B'$ still has this property). The relation between these two notions stems from the fact that also for universal submersions there exists a criterion in terms of discrete valuation domains: A morphism of finite type between two affine noetherian schemes is a universal submersion if and only if the base change to any discrete valuation domain yields a submersion. For a morphism

$Z\longrightarrow \operatorname {Spec} {\left(D\right)}$

(${}D$ a discrete valuation domain) to be a submersion means that above the only chain of prime ideals in ${}\operatorname {Spec} {\left(D\right)}$, namely ${}(0)\subset {\mathfrak {m}}_{D}$, there exists a chain of prime ideals ${}{\mathfrak {p}}'\subseteq {\mathfrak {q}}'$ in ${}Z$ lying over this chain. This pair-lifting property holds for a universal submersion

$\operatorname {Spec} {\left(S\right)}\longrightarrow \operatorname {Spec} {\left(R\right)}$

for any pair of prime ideals ${}{\mathfrak {p}}\subseteq {\mathfrak {q}}$ in ${}\operatorname {Spec} {\left(R\right)}$. This property is stronger than lying over (which means surjective) but weaker than the going-down or the going-up property (in the presence of surjectivity).

If we are dealing only with algebras of finite type over the complex numbers ${}{\mathbb {C} }$, then we may also consider the corresponding complex spaces with their natural topology induced from the euclidean topology of ${}{\mathbb {C} }^{n}$. Then universal submersive with respect to the Zariski topology is the same as submersive in the complex topology (the target space needs to be normal).

Example

Let ${}K$ be a field and consider ${}R=K[X]$. Since this is a principal ideal domain, the only interesting forcing algebras (if we are only interested in the local behavior around ${}(X)$) are of the form ${}K[X,T]/{\left(X^{n}T-X^{m}\right)}$. For ${}m\geq n$ this ${}K[X]$-algebra admits a section (corresponding to the fact that ${}X^{m}\in {\left(X^{n}\right)}$), and if ${}n\geq 1$ there exists an affine line over the maximal ideal ${}(X)$. So now assume ${}m<n$. If ${}m\geq 0$ then we have a hyperbola mapping to an affine line, with the fiber over ${}(X)$ being empty, corresponding to the fact that ${}1$ does not belong to the radical of ${}{\left(X^{n}\right)}$ for ${}n\geq 1$. So assume finally ${}1\leq m<n$. Then ${}X^{m}$ belongs to the radical of ${}{\left(X^{n}\right)}$, but not to its integral closure (which is the identical closure on a one-dimensional regular ring). We can write the forcing equation as ${}X^{n}T-X^{m}=X^{m}{\left(X^{n-m}T-1\right)}$. So the spectrum of the forcing algebra consists of a (thickend) line over ${}(X)$ and of a hyperbola. The forcing morphism is surjective, but it is not a submersion. For example, the preimage of ${}V(X)=\{(X)\}$ is a connected component hence open, but this single point is not open.

Example

Let ${}K$ be a field and let ${}R=K[X,Y]$ be the polynomial ring in two variables. We consider the ideal ${}I={\left(X^{2},Y\right)}$ and the element ${}X$. This element belongs to the radical of this ideal, hence the forcing morphism

$\operatorname {Spec} {\left(K[X,Y,T_{1},T_{2}]/{\left(X^{2}T_{1}+YT_{2}+X\right)}\right)}\longrightarrow \operatorname {Spec} {\left(K[X,Y]\right)}$

is surjective. We claim that it is not a submersion. For this we look at the reduction modulo ${}Y$. In ${}K[X,Y]/(Y)\cong K[X]$ the ideal ${}I$ becomes ${}{\left(X^{2}\right)}$ which does not contain ${}X$. Hence by the valuative criterion for integral closure, ${}X$ does not belong to the integral closure of the ideal. One can also say that the chain ${}V(X,Y)\subset V(Y)$ in the affine plane does not have a lift (as a chain) to the spectrum of the forcing algebra.

For the ideal

${}I={\left(X^{2},Y^{2}\right)}\,$

and the element ${}XY$ the situation looks different. Let

$\theta \colon K[X,Y]\longrightarrow D$

be a ring homomorphism to a discrete valuation domain ${}D$. If ${}X$ or ${}Y$ is mapped to ${}0$, then also ${}XY$ is mapped to ${}0$ and hence belongs to the extended ideal. So assume that ${}\theta (X)=u\pi ^{r}$ and ${}\theta (Y)=v\pi ^{s}$, where ${}\pi$ is a local parameter of ${}D$ and ${}u$ and ${}v$ are units. Then ${}\theta (XY)=uv\pi ^{r+s}$ and the exponent is at least the minimum of ${}2r$ and ${}2s$, hence

${}\theta (XY)\in {\left(\pi ^{2r},\pi ^{2s}\right)}={\left(\theta {\left(X^{2}\right)},\theta {\left(Y^{2}\right)}\right)}D\,.$

So ${}XY$ belongs to the integral closure of ${}{\left(X^{2},Y^{2}\right)}$ and the forcing morphism

$\operatorname {Spec} {\left(K[X,Y,T_{1},T_{2}]/{\left(X^{2}T_{1}+Y^{2}T_{2}+XY\right)}\right)}\longrightarrow \operatorname {Spec} {\left(K[X,Y]\right)}$

is a universal submersion.

Continuous closure

Suppose now that ${}R={\mathbb {C} }[X_{1},\ldots ,X_{k}]$. Then every polynomial ${}f\in R$ can be considered as a continuous function

$f\colon {\mathbb {C} }^{k}\longrightarrow {\mathbb {C} },\,{\left(x_{1},\ldots ,x_{k}\right)}\longmapsto f{\left(x_{1},\ldots ,x_{k}\right)},$

in the complex topology. If ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ is an ideal and ${}f\in R$ is an element, we say that ${}f$ belongs to the continuous closure of ${}I$, if there exist continuous functions

$g_{1},\ldots ,g_{n}\colon {\mathbb {C} }^{k}\longrightarrow {\mathbb {C} }$

such that

${}f=\sum _{i=1}^{n}g_{i}f_{i}\,$

(as an identity of functions). The same definition works for ${}{\mathbb {C} }$-algebras of finite type.

It is not at all clear at once that there may exist polynomials ${}f\notin I$ but inside the continuous closure of ${}I$. For ${}{\mathbb {C} }[X]$ it is easy to show that the continuous closure is (like the integral closure) just the ideal itself. We also remark that when we would only allow holomorphic functions ${}g_{1},\ldots ,g_{n}$ then we could not get something larger. However, with continuous functions

$g_{1},g_{2}\colon {\mathbb {C} }^{2}\longrightarrow {\mathbb {C} }$

we can for example write

${}X^{2}Y^{2}=g_{1}X^{3}+g_{2}Y^{3}\,.$

Continuous closure is always inside the integral closure and hence also inside the radical. The element ${}XY$ does not belong to the continuous closure of ${}I=(X^{2},Y^{2})$, though it belongs to the integral closure of ${}I$. In terms of forcing algebras, an element ${}f$ belongs to the continuous closure if and only if the complex forcing mapping

$\varphi _{\mathbb {C} }\colon \operatorname {Spec} {\left(B\right)}_{\mathbb {C} }\longrightarrow \operatorname {Spec} {\left(R\right)}_{\mathbb {C} }$

(between the corresponding complex spaces) admits a continuous section.

Vector bundles and torsors

Geometric vector bundles

We have seen that the fibers of the spectrum of a forcing algebra are (empty or) affine spaces. However, this is not only fiberwise true, but more general: If we localize the forcing algebra at ${}f_{i}$ we get

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

since we can write

${}T_{i}=-\sum _{j\neq i}{\frac {f_{j}}{f_{i}}}T_{j}+{\frac {f}{f_{i}}}\,.$

So over every ${}D(f_{i})$ the spectrum of the forcing algebra is an ${}(n-1)$-dimensional affine space over the base. So locally, restricted to ${}D(f_{i})$, we have isomorphisms

${}T{|}_{D(f_{i})}\cong D(f_{i})\times {{\mathbb {A} }_{}^{n-1}}\,.$

On the intersections ${}D(f_{i})\cap D(f_{j})$ we get two identifications with affine space, and the transition morphisms are linear if ${}f=0$, but only affine-linear in general (because of the translation with ${\frac {f}{f_{i}}}$).

So 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 ${}D(f_{i})\times {{\mathbb {A} }_{}^{n-1}}$. 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 ${}V{|}_{U}$, where ${}V$ is the spectrum of a homogeneous forcing algebra, 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 ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ideal. Then on ${}U=D(I)$ we have the short exact sequence

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

On the left we have a locally free sheaf of rank ${}n-1$ which we call the syzygy sheaf or syzygy bundle. It is the sheaf of sections in the geometric vector bundle ${}\operatorname {Spec} {\left(R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}\right)}\right)}{|}_{U}$.  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.

Tight closure and cohomological properties of torsors

The closure operations we have considered so far 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

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

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.

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.

Tight closure and solid 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 ${}zf^{q}\in {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}$, 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 ${}H^{d-1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0$ then this is true for all quasicoherent sheaves instead of 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 3.1 - 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.

An important property of tight closure is that it is trivial for regular rings, i.e.

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

for every ideal ${}I$. This rests upon Kunz's theorem saying that the Frobenius homomorphism for regular rings is flat. This property implies the following cohomological property of torsors.

Corollary

Let ${}(R,{\mathfrak {m}})$ denote a regular local ring of dimension ${}d$ and of positive characteristic, let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ${}{\mathfrak {m}}$-primary ideal and ${}f\in R$ be an element with ${}f\notin I$. Let ${}B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}$ be the corresponding forcing algebra.

Then the extended ideal ${}{\mathfrak {m}}B$ satisfies

${}H_{{\mathfrak {m}}B}^{d}(B)=H^{d-1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0\,.$

Proof

This follows from Theorem 3.1 and ${}f\notin I^{*}$.

$\Box$

In dimension two this is true in every (even mixed) characteristic.

Theorem

Let ${}(R,{\mathfrak {m}})$ denote a two-dimensional regular local ring, let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ${}{\mathfrak {m}}$-primary ideal and ${}f\in R$ an element with ${}f\notin I$. Let

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

be the corresponding forcing algebra.

Then for the extended ideal ${}{\mathfrak {m}}B$ we have

${}H_{{\mathfrak {m}}B}^{2}(B)=H^{1}(D({\mathfrak {m}}B),{\mathcal {O}}_{B})=0\,.$

In particular, the open subset ${}T=D({\mathfrak {m}}B)$ is an affine scheme if and only if ${}f\notin I$.

The main point for the proof of this result is that for ${}f\notin I$, the natural mapping

$H^{1}(U,{\mathcal {O}}_{X})\longrightarrow H^{1}(T,{\mathcal {O}}_{T})$

is not injective by a Matlis duality argument. Since the local cohomology of a regular ring is explicitly known, this map annihilates some cohomology class of the form ${}{\frac {1}{fg}}$ where ${}f,g$ are parameters. But then it annihilates the complete local cohomology module and then ${}T$ is an affine scheme.

For non-regular two-dimensional rings it is a difficult question in general to decide whether a torsor is affine or not. A satisfactory answer is only known in the normal two-dimensional graded case over a field, which we will deal with in the final lectures.

In higher dimension in characteristic zero it is not true that a regular ring is solidly closed (meaning that every ideal equals its solid closure), as was shown by the following example of Paul Roberts.

Example

Let ${}K$ be a field of characteristic ${}0$ and let

${}B=K[X,Y,Z][U,V,W]/{\left(X^{3}U+Y^{3}V+Z^{3}W-X^{2}Y^{2}Z^{2}\right)}\,.$

Then the ideal ${}{\mathfrak {a}}=(X,Y,Z)B$ has the property that ${}H_{\mathfrak {a}}^{3}(B)\neq 0$. This means that in ${}R=K[X,Y,Z]$, the element ${}X^{2}Y^{2}Z^{2}$ belongs to the solid closure of the ideal ${}{\left(X^{3},Y^{3},Z^{3}\right)}$, and hence the three-dimensional polynomial ring is not solidly closed.

This example was the motivation for the introduction of parasolid closure, which has all the good properties of solid closure but which is also trivial for regular rings.

If ${}R$ is a normal local domain of dimension ${}2$ and ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ an ${}{\mathfrak {m}}$-primary ideal, then ${}f\in I^{*}$ (or inside the solid closure) if and only if ${}D({\mathfrak {m}})\subseteq \operatorname {Spec} {\left(B\right)}$ is not an affine scheme, where ${}B$ denotes the forcing algebra. Here we will discuss in more detail, with this application in mind, when a scheme is affine.

Affine schemes

A scheme ${}U$ is called affine if it is isomorphic to the spectrum of some commutative ring ${}R$. If the scheme is of finite type over a field (or a ring) ${}K$ (if we have a variety), then this is equivalent to saying that there exist global functions

${}g_{1},\ldots ,g_{m}\in \Gamma (U,{\mathcal {O}}_{U})\,$

such that the mapping

$U\longrightarrow {{\mathbb {A} }_{K}^{m}},\,x\longmapsto {\left(g_{1}(x),\ldots ,g_{m}(x)\right)},$

is a closed embedding. The relation to cohomology is given by the following well-known theorem of Serre.

Theorem

Let ${}U$ denote a noetherian scheme. Then the following properties are equivalent.

1. ${}U$ is an affine scheme.
2. For every quasicoherent sheaf ${}{\mathcal {F}}$ on ${}U$ and all ${}i\geq 1$ we have ${}H^{i}(U,{\mathcal {F}})=0$.
3. For every coherent ideal sheaf ${}{\mathcal {I}}$ on ${}U$ we have ${}H^{1}(U,{\mathcal {I}})=0$.

It is in general a difficult question whether a given scheme ${}U$ is affine. For example, suppose that ${}X=\operatorname {Spec} {\left(R\right)}$ is an affine scheme and

${}U=D({\mathfrak {a}})\subseteq X\,$

is an open subset (such schemes are called quasiaffine) defined by an ideal ${}{\mathfrak {a}}\subseteq R$. When is ${}U$ itself affine? The cohomological criterion above simplifies to the condition that ${}H^{i}(U,{\mathcal {O}}_{X})=0$ for ${}i\geq 1$.

Of course, if ${}{\mathfrak {a}}=(f)$ is a principal ideal (or up to radical a principal ideal), then

${}U=D(f)\cong \operatorname {Spec} {\left(R_{f}\right)}\,$

is affine. On the other hand, if ${}(R,{\mathfrak {m}})$ is a local ring of dimension ${}\geq 2$, then

${}D({\mathfrak {m}})\subset \operatorname {Spec} {\left(R\right)}\,$

is not affine, since

${}H^{d-1}(U,{\mathcal {O}}_{X})=H_{\mathfrak {m}}^{d}(R)\neq 0\,$

by the relation between sheaf cohomology and local cohomology and a Theorem of Grothendieck.

Codimension condition

One can show that for an open affine subset ${}U\subseteq X$ the closed complement ${}Y=X\setminus U$ must be of pure codimension one (${}U$ must be the complement of the support of an effective divisor). In a regular or (locally ${}\mathbb {Q}$)- factorial domain the complement of every effective divisor is affine, since the divisor can be described (at least locally geometrically) by one equation. But it is easy to give examples to show that this is not true for normal three-dimensional domains. The following example is a standard example for this phenomenon and it is in fact given by a forcing algebra.

Example

Let ${}K$ be a field and consider the ring

${}R=K[x,y,u,v]/(xu-yv)\,.$

The ideal ${}{\mathfrak {p}}=(x,y)$ is a prime ideal in ${}R$ of height one. Hence the open subset ${}U=D(x,y)$ is the complement of an irreducible hypersurface. However, ${}U$ is not affine. For this we consider the closed subscheme

${}{\mathbb {A} }_{K}^{2}\cong Z=V(u,v)\subseteq \operatorname {Spec} {\left(R\right)}\,$

and ${}Z\cap U\subseteq U$. If ${}U$ were affine, then also the closed subscheme ${}Z\cap U\cong {\mathbb {A} }_{K}^{2}\setminus \{(0,0)\}$ would be affine, but this is not true, since the complement of the punctured plane has codimension ${}2$.

Ring of global sections of affine schemes

For an open subset ${}U=D({\mathfrak {a}})\subseteq \operatorname {Spec} {\left(R\right)}$ its ring of global sections ${}\Gamma (U,{\mathcal {O}}_{X})$ is difficult to compute in general. If ${}R$ is a domain and ${}{\mathfrak {a}}={\left(f_{1},\ldots ,f_{n}\right)}$, then

${}\Gamma (U,{\mathcal {O}}_{X})=R_{f_{1}}\cap R_{f_{2}}\cap \ldots \cap R_{f_{n}}\,.$

This ring is not always of finite type over ${}R$, but it is if ${}U$ is affine.

Lemma

Let ${}R$ be a noetherian ring and ${}U=D({\mathfrak {a}})\subseteq \operatorname {Spec} {\left(R\right)}$ an open subset.

Then the following hold.

1. ${}U$ is an affine scheme if and only if ${}{\mathfrak {a}}\Gamma (U,{\mathcal {O}}_{X})=(1)$.
2. If this holds, and ${}q_{1}f_{1}+\cdots +q_{n}f_{n}=1$ with ${}f_{1},\ldots ,f_{n}\in {\mathfrak {a}}$ and ${}q_{i}\in \Gamma (U,{\mathcal {O}}_{X})$, then ${}\Gamma (U,{\mathcal {O}}_{X})=R[q_{1},\ldots ,q_{n}]$. In particular, the ring of global sections over ${}U$ is finitely generated over ${}R$.

Proof

We only give a sketch. (1). There always exists a natural scheme morphism

$U\longrightarrow \operatorname {Spec} {\left(\Gamma (U,{\mathcal {O}}_{X})\right)},$

and ${}U$ is affine if and only if this morphism is an isomorphism. It is always an open embedding (because it is an isomorphism on the $D(f)$, ${}f\in {\mathfrak {a}}$), and the image is ${}D({\mathfrak {a}}\Gamma (U,{\mathcal {O}}_{X}))$. This is everything if and only if the extended ideal is the unit ideal.

(2). We write ${}1=q_{1}f_{1}+\cdots +q_{n}f_{n}$ and consider the natural morphism

$U\longrightarrow \operatorname {Spec} {\left(R[q_{1},\ldots ,q_{n}]\right)}$

corresponding to the ring inclusion ${}R[q_{1},\ldots ,q_{n}]\subseteq \Gamma (U,{\mathcal {O}}_{X})$. This morphism is again an open embedding and its image is everything.

$\Box$

We give some examples of tight closure computations on the Fermat cubic ${}x^{3}+y^{3}+z^{3}=0$, a standard example in tight closure theory, with the methods we have developed so far.

Example

We consider the Fermat cubic ${}R=K[X,Y,Z]/{\left(X^{3}+Y^{3}+Z^{3}\right)}$, the ideal ${}I=(X,Y)$ and the element ${}Z$. We claim that for characteristic ${}\neq 3$ the element ${}Z$ does not belong to the solid closure of ${}I$. Equivalently, the open subset

${}D(X,Y)\subseteq \operatorname {Spec} {\left(R[S,T]/(XS+YT-Z)\right)}\,$

is affine. For this we show that the extended ideal inside the ring of global sections is the unit ideal. First of all we get the equation

${}X^{3}+Y^{3}=(XS+YT)^{3}=X^{3}S^{3}+3X^{2}S^{2}YT+3XSY^{2}T^{2}+Y^{3}T^{3}\,$

or, equivalently,

${}X^{3}{\left(S^{3}-1\right)}+3X^{2}YS^{2}T+3XY^{2}ST^{2}+Y^{3}{\left(T^{3}-1\right)}=0\,.$

We write this as

${}{\begin{aligned}X^{3}{\left(S^{3}-1\right)}&=-3X^{2}YS^{2}T-3XY^{2}ST^{2}-Y^{3}{\left(T^{3}-1\right)}\\&=Y{\left(-3X^{2}S^{2}T-3XYST^{2}-Y^{2}{\left(T^{3}-1\right)}\right)},\end{aligned}}$

which yields on ${}D(X,Y)$ the rational function

${}Q={\frac {S^{3}-1}{Y}}={\frac {-3X^{2}S^{2}T-3XYST^{2}-Y^{2}{\left(T^{3}-1\right)}}{X^{3}}}\,.$

This shows that ${}S^{3}-1=QY$ belongs to the extended ideal. Similarly, one can show that also the other coefficients ${}3S^{2}T,3ST^{2},T^{3}-1$ belong to the extended ideal. Therefore in characteristic different from ${}3$, the extended ideal is the unit ideal.

Example

We consider the Fermat cubic ${}R=K[X,Y,Z]/{\left(X^{3}+Y^{3}+Z^{3}\right)}$, the ideal ${}I=(X,Y)$ and the element ${}Z^{2}$. We claim that in positive characteristic ${}\neq 3$ the element ${}Z^{2}$ does belong to the tight closure of ${}I$. Equivalently, the open subset

${}D(X,Y)\subseteq \operatorname {Spec} {\left(R[S,T]/(XS+YT+Z^{2})\right)}\,$

is not affine. The element ${}Z^{2}$ defines the cohomology class

${}c={\frac {Z^{2}}{XY}}\in H^{1}(D(X,Y),{\mathcal {O}}_{X})\,$

and its Frobenius pull-backs are

${}F^{e*}(c)={\frac {Z^{2q}}{X^{q}Y^{q}}}\in H^{1}(D(X,Y),{\mathcal {O}}_{X})\,.$

This cohomology module has a ${}\mathbb {Z}$-graded structure (the degree is given by the difference of the degree of the numerator and the degree of the denominator) and, moreover, it is ${}0$ in positive degree (this is related to the fact that the corresponding projective curve is elliptic). Therefore for any homogeneous element ${}t\in R$ of positive degree we have ${}tF^{e*}(c)=0$ and so ${}Z^{2}$ belongs to the tight closure.

From this it follows also that in characteristic ${}0$ the element ${}Z^{2}$ belongs to the solid closure, because affineness is an open property in an arithmetic family.

We give now a cohomological proof of a tight closure containment on the Fermat cubic for a non-parameter ideal. M. McDermott has raised the question whether

$xyz\in (x^{2},y^{2},z^{2})^{*}{\text{ in }}K[x,y,z]/(x^{3}+y^{3}+z^{3}).$

This was answered positively by A. Singh by a long “equational” argument.

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 ${}f=xyz$. We consider the short exact sequence

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

and the cohomology class

${}c=\delta (xyz)=H^{1}{\left(U,\operatorname {Syz} {\left(x^{2},y^{2},z^{2}\right)}\right)}\,.$

We want to show that ${}zF^{e*}(c)=0$ for all ${}e\geq 0$ (here the test element ${}z$ equals the element ${}z$ in the ring). It is helpful to work with the graded structure on this syzygy sheaf (or to work on the corresponding elliptic curve ${}\operatorname {Proj} R$ directly). Now the equation ${}x^{3}+y^{3}+z^{3}=0$ can be considered as a syzygy (of total degree ${}3$) for ${}x^{2},y^{2},z^{2}$, yielding an inclusion

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

Since this syzygy does not vanish anywhere on ${}U$ the quotient sheaf is invertible and in fact isomorphic to the structure sheaf. Hence we have

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

and the cohomology sequence

$\longrightarrow H^{1}{\left(U,{\mathcal {O}}_{U}\right)}_{s}\longrightarrow H^{1}{\left(U,\operatorname {Syz} {\left(x^{2},y^{2},z^{2}\right)}\right)}_{s+3}\longrightarrow H^{1}{\left(U,{\mathcal {O}}_{U}\right)}_{s}\longrightarrow 0,$

where ${}s$ denotes the degree-${}s$th piece. Our cohomology class ${}c$ lives in ${}H^{1}{\left(U,\operatorname {Syz} {\left(x^{2},y^{2},z^{2}\right)}\right)}_{3}$, so its Frobenius pull-backs live in ${}H^{1}{\left(U,F^{e*}\operatorname {Syz} {\left(x^{2},y^{2},z^{2}\right)}\right)}_{3q}$, and we can have a look at the cohomology of the pull-backs of the sequence, i.e.

$\longrightarrow H^{1}{\left(U,{\mathcal {O}}_{U}\right)}_{0}\longrightarrow H^{1}{\left(U,F^{e*}\operatorname {Syz} {\left(x^{2},y^{2},z^{2}\right)}\right)}_{3q}\longrightarrow H^{1}{\left(U,{\mathcal {O}}_{U}\right)}_{0}\longrightarrow 0.$

The class ${}zF^{e*}(c)$ lives in ${}H^{1}{\left(U,F^{e*}\operatorname {Syz} {\left(x^{2},y^{2},z^{2}\right)}\right)}_{3q+1}$. It is mapped on the right to ${}H^{1}{\left(U,{\mathcal {O}}_{U}\right)}_{1}$, which is ${}0$ (because we are working over an elliptic curve), hence it comes from the left, which is

${}H^{1}(U,{\mathcal {O}}_{U})_{1}=0\,.$

So ${}zF^{e*}(c)=0$ and ${}f\in {\left(x^{2},y^{2},z^{2}\right)}^{*}$.

Affineness and superheight

We have mentioned above that the complement of an affine open subset must have pure codimension ${}1$. We have also seen in Example 3.6 that the non-affineness can be established by looking at the behavior of the codimension when the situation is restricted to closed subschemes. The following definition and theorem is an algebraic version of this observation.

Definition

Let ${}R$ be a noetherian commutative ring and let ${}I\subseteq R$ be an ideal. The (noetherian) superheight is the supremum

$\operatorname {sup} _{}^{}{\left(\operatorname {ht} _{}^{}{\left(IS\right)}:\,S{\text{ is a notherian }}R-{\text{algebra}}\right)}.$

Theorem

Let ${}R$ be a noetherian commutative ring and let ${}I\subseteq R$ be an ideal and ${}U=D(I)\subseteq X=\operatorname {Spec} {\left(R\right)}$. Then the following are equivalent.

1. ${}U$ is an affine scheme.
2. ${}I$ has superheight ${}\leq 1$ and ${}\Gamma (U,{\mathcal {O}}_{X})$ is a finitely generated ${}R$-algebra.

It is not true at all that the ring of global sections of an open subset ${}U$ of the spectrum ${}X$ of a noetherian ring is of finite type over this ring. This is not even true if ${}X$ is an affine variety. This problem is directly related to Hilbert's fourteenth problem, which has a negative answer. We will later present examples where ${}U$ has superheight one, yet is not affine, hence its ring of global sections is not finitely generated.

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 torsor which meets the exceptional fiber (the fiber over the maximal ideal) in isolated points, and this 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.

In characteristic zero, the plus closure behaves very differently compared with positive characteristic. If ${}R$ is a normal domain of characteristic ${}0$, then the trace map shows that the plus closure is trivial, ${}I^{+}=I$ for every ideal ${}I$.

Examples

In the following two examples we use results from tight closure theory to establish (non)-affineness properties of certain torsors.

Example

Let ${}K$ be a field and consider the Fermat ring

${}R=K[X,Y,Z]/{\left(X^{d}+Y^{d}+Z^{d}\right)}\,$

together with the ideal ${}I=(X,Y)$ and ${}f=Z^{2}$. For ${}d\geq 3$ we have ${}Z^{2}\notin (X,Y)$. This element is however in the tight closure ${}(X,Y)^{*}$ of the ideal in positive characteristic (assume that the characteristic ${}p$ does not divide ${}d$) and is therefore also in characteristic ${}0$ inside the tight closure and inside the solid closure. Hence the open subset

${}D(X,Y)\subseteq \operatorname {Spec} {\left(K[X,Y,Z,S,T]/{\left(X^{d}+Y^{d}+Z^{d},SX+TY-Z^{2}\right)}\right)}\,$

is not an affine scheme. In positive characteristic, ${}Z^{2}$ is also contained in the plus closure ${}(X,Y)^{+}$ and therefore this open subset contains punctured surfaces (the spectrum of the forcing algebra contains two-dimensional closed subschemes which meet the exceptional fiber ${}V(X,Y)$ in only one point; the ideal ${}(X,Y)$ has superheight two in the forcing algebra). In characteristic zero however, the superheight is one because plus closure is trivial for normal domains in characteristic ${}0$, and therefore by Theorem 3.12 the algebra ${}\Gamma ({D(X,Y),{\mathcal {O}}}_{B})$ is not finitely generated. For ${}K={\mathbb {C} }$ and ${}d=3$ one can also show that ${}D(X,Y)_{\mathbb {C} }$ is, considered as a complex space, a Stein space.

Example

Let ${}K$ be a field of positive characteristic ${}p\geq 7$ and consider the ring

${}R=K[X,Y,Z]/{\left(X^{5}+Y^{3}+Z^{2}\right)}\,$

together with the ideal ${}I=(X,Y)$ and ${}f=Z$. Since ${}R$ has a rational singularity, it is ${}F$-regular, i.e. all ideals are tightly closed. Therefore ${}Z\notin (X,Y)^{*}$ and so the torsor

${}D(X,Y)\subseteq \operatorname {Spec} {\left(K[X,Y,Z,S,T]/{\left(X^{5}+Y^{3}+Z^{2},SX+TY-Z\right)}\right)}\,$

is an affine scheme. In characteristic zero this can be proved by either using that ${}R$ is a quotient singularity or by using the natural grading ($\operatorname {deg} _{}^{}{\left(X\right)}=6,\,\operatorname {deg} _{}^{}{\left(Y\right)}=10,\,\operatorname {deg} _{}^{}{\left(Z\right)}=15$) where the corresponding cohomology class ${}{\frac {Z}{XY}}$ gets degree ${}-1$ and then applying the geometric criteria (see below) on the corresponding projective curve (rather the corresponding curve of the standard-homogenization ${}U^{30}+V^{30}+W^{30}=0$).

Cones over projective curves

We continue with the question when the torsors given by a forcing algebra over a two-dimensional ring are affine? We will look at the graded situation to be able to work on the corresponding projective curve.

In particular we want to address the following questions:

1. Is there a procedure to decide whether the torsor is affine?
2. Is it non-affine if and only if there exists a geometric reason for it not to be affine (because the superheight is too large)?
3. How does the affineness vary in an arithmetic family, when we vary the prime characteristic?
4. How does the affineness vary in a geometric family, when we vary the base ring?

In terms of tight closure, these questions are directly related to the tantalizing question of tight closure (is it the same as plus closure), the dependence of tight closure on the characteristic and the localization problem of tight closure.

Geometric interpretation in dimension two

We will restrict now to the two-dimensional homogeneous case in order to work on the corresponding projective curve. We want to find an object over the curve which corresponds to the forcing algebra or its induced torsor.

Let ${}R$ be a two-dimensional standard-graded normal domain over an algebraically closed field ${}K$. Let ${}C=\operatorname {Proj} {\left(R\right)}$ be the corresponding smooth projective curve 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 ${}C$ the short exact sequence

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

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

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

Recall that the degree of a vector bundle ${}{\mathcal {S}}$ on a projective curve is defined as the degree of the invertible sheaf ${}\bigwedge ^{r}{\mathcal {S}}$, where ${}r$ is the rank of $$