# Kurs:Vector bundles and ideal closure operations (MSRI 2012)/Lecture 1

*Systems of linear equations*

We start with some linear algebra. Let be a field. We consider a system of linear homogeneous equations over ,

where the are elements in . The solution set to this system of homogeneous equations is a vector space over (a subvector space of ), its dimension is , where

is the matrix given by these elements. Additional elements give rise to the system of inhomogeneous linear equations,

The solution set 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

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 on the set . Moreover, if we fix one solution

(supposing that at least one solution exists), then there exists a bijection

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

Suppose now that 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 we have functions

on (which are continuous, or differentiable, or algebraic). We form the matrix of functions , which yields for every point a matrix over . Then we get from these data the space

together with the projection to . For a fixed point , the fiber of over is the solution space to the corresponding system of homogeneous linear equations given by inserting into . In particular, all fibers of the map

are vector spaces
(maybe of non-constant dimension).
These vector space structures yield an addition^{[1]}

(only points in the same fiber can be added). The mapping

is called the *zero-section*.

Suppose now that additional functions

are given. Then we can form the set

with the projection to . Again, every fiber of over a point is the solution set to the system of inhomogeneous linear equations which arises by inserting into and . The actions of the fibers on (coming from linear algebra) extend to an action

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

with exists, then we can construct a (continuous, differentiable, algebraic) isomorphism between and . However, different from the situation in linear algebra (which corresponds to the situation where is just one point), such a section does rarely exist.

These objects 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

is given. Then in the homogeneous case () the fibers are vector spaces of dimension or , and the later holds exactly for the points where . In the inhomogeneous case the fibers are either empty or of dimension or . We give some typical examples.

We consider the line (or etc.) with the (identical) function . For and , i.e. for the homogeneous equation , the geometric object consists of a horizontal line (corresponding to the zero-solution) and a vertical line over . So all fibers except one are zero-dimensional vector spaces. For the inhomogeneous equation , is a hyperbola, and all fibers are zero-dimensional with the exception that the fiber over is empty.

For the homogeneous equation , is just the affine cylinder over the base line. For the inhomogeneous equation , consists of one vertical line, almost all fibers are empty.

Let denote a plane (like ) with coordinate functions and . We consider an inhomogeneous linear equation of type

The fiber of the solution set over a point is one-dimensional, whereas the fiber over has dimension two (for ). Many properties of depend on these four exponents.

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

*Forcing algebras*

We describe now the algebraic setting of systems of inhomogeneous linear equations depending on a base space. For a commutative ring , its spectrum is a topological space on which the ring elements can be considered as functions. The value of at a prime ideal is just the image of under the morphism . In this interpretation, a ring element is a function with values in different fields. Suppose that contains a field . Then an element gives rise to the ring homomorphism

which itself gives rise to a scheme morphism

This is another way to consider as a function on .

The following object was introduced by M. Hochster in 1994 in his work on solid closure.

Let be a commutative ring and let and be elements in . Then the -algebra

is called the *forcing algebra* of these elements
(or these data).

The forcing algebra forces to lie inside the extended ideal (hence the name) For every -algebra such that there exists a (non unique) ring homomorphism by sending to the coefficient in an expression .

The forcing algebra induces the spectrum morphism

Over a point , the fiber of this morphism is given by

and we can write

where means the evaluation of the in the residue class field. Hence the -points in the fiber are exactly the solutions to the inhomogeneous linear equation . In particular, all the fibers are (empty or) affine spaces.

As the solution vector space of a system of homogeneous linear equations acts on the solution set of a system of inhomogeneous linear equations, the spectrum of a homogeneous forcing algebra acts on the spectrum of an inhomogeneous forcing algebra. The spectrum

is a group scheme over , and the Hopfalgebra structure is given by . Similarly, the action of on is given by

On the ring level this map is again induced by .

*Forcing algebras and closure operations*

Let denote a commutative ring and let be an ideal. Let and let

be the corresponding forcing algebra and

the corresponding spectrum morphism. How are properties of (or of the -algebra ) related to certain ideal closure operations?

We start with some examples. The element belongs to the ideal if and only if we can write . By the universal property of the forcing algebra this means that there exists an -algebra-homomorphism

hence holds if and only if admits a scheme section. This is also equivalent to

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

We have a look at the radical of the ideal ,

As this is quite a coarse closure operation we should expect that this corresponds to a quite coarse property of the morphism as well. Indeed, it is true that if and only if is surjective. This and the interpretation of other closure operations like integral closure and in particular tight closure in terms of forcing algebras will be discussed in the tutorial session and in the next lectures.

*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 fiberwisely true, but more generally: If we localize the forcing algebra at we get

On the intersections we get two identifications with affine space, and the transition morphisms are linear if , but only affine-linear in general (because of the translation with ). This local description holds on the union . Moreover, in the homogeneous case () the transition mappings are linear. Hence is a geometric vector bundle according to the following definition.

Let denote a scheme. A scheme equipped with a morphism

is called a
*geometric vector bundle*
of rank over if there exists an open covering
and -isomorphisms

such that for every open affine subset the transition mappings

are linear automorphisms, i.e. they are induced by an automorphism of the polynomial ring given by .

Here we can always restrict to affine open coverings. If is separated then the intersection of two affine open subschemes is again affine and then it is enough to check the condition on the intersection. The trivial bundle of rank is the -dimensional affine space over , 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

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

The reason for this is that the isomorphisms occurring in the definition of a geometric vector bundle are linear, hence the addition on coming from an isomorphism with some affine space over is independent of the choosen isomorphism. For the same reason there is a unique closed subscheme of called the *zero-section* which is locally defined to be
.
Also, the multiplication by a scalar, i.e. the mapping

carries over to a scalar multiplication

In particular, for every point the fiber is an affine space over .

For a geometric vector bundle and an open subset one sets

so this is the set of sections in over . This gives in fact for every scheme over 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.

A coherent -module on a scheme is called
*locally free*
of rank , if there exists an open covering
and -module-isomorphisms
for every
.

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

Let denote a scheme.

** Then the category of locally free sheaves on and the category of geometric vector bundles on are equivalent. **

A geometric vector bundle corresponds to the sheaf of its sections, and a locally free sheaf corresponds to the (relative) spectrum of the symmetric algebra of the dual module .

The free sheaf of rank corresponds to the affine space over .

The global sections in

So these are just the syzygies for the ideal generators, and they form the syzygy module. We denote this by . The sheaf of sections in is also the sheafification of this syzygy module. The restriction of this sheaf to is locally free and corresponds to the geometric vector bundle .

This action of on induces an action of the vector bundle on , and endowed with this action becomes a torsor. We will deal with this structure in the next lecture.

*Fußnoten*

- ↑ is the fiber product of with itself.