# Kurs:Vector bundles and their torsors (Kolkata 2011)/Lecture 1

*Locally free sheaves*

We start this lecture series by asking what are the easiest modules over a commutative ring . There are several possible answers to this question, but the answer should definitely include the ring itself and also the -module. Another easy module is the direct product of the ring with itself. These modules are called free modules of rank . Ideals might look easy at first sight, but in fact they are not, with the exception of a principal ideal domain, where all non-zero ideals are isomorphic as a module to . Instead we consider here -modules which have the property that their localizations are free. For this we look at a typical example, the so-called syzygy modules.

Let be a commutative ring and let be an ideal generated by finitely many elements . The free resolution of the residue class ring is the exact complex

This resolution goes (unless has finite projective dimension) on forever, but we can break it up to obtain the exact complex

where the module is just defined to be the kernel of the -module-homomorphism

This kernel consists exactly of the syzygies for these elements, hence it is called (the first) syzygy module. This module can be already quite complicated, however, we can make the following observation. Let us fix one , say , and look at the induced sequence over the localization . As localization is an exact functor, we still get an exact sequence, and since , the ideal contains now a unit and therefore we have , so we can rewrite the induced sequence as

We claim that we have an -module isomorphism

by sending the -th standard vector () to

(the stands at the th position). This is obviously well-defined, since is a unit in , and evidently the given tuple is a syzygy. If is a syzygy, then is a preimage, since it is mapped under this homomorphism to

Hence we have a surjection. The injectivity follows immediately by looking at the components to in the syzygy.

This means that the syzygy module when restricted to the open subset (viewed as an -module) is free of rank , and the same holds for all . Hence the syzygy module restricted to the open subset

has the property that there exists a covering by open subsets such that the restrictions to these open subsets are free modules. In general, the syzygy module is not free as an -module nor as an -module on . The above given explicit isomorphism on (such an isomorphism is called a local trivialization of on ) uses that is a unit, hence this can not be extended to give an isomorphism on .

On the intersection as well as are units, hence the above isomorphisms (let's call them on and on ) induce two different isomorphisms on between and We can connect them to get an isomorphism

which is given by the (over ) invertible -matrix

We have seen that a syzygy module as above considered on has the following two properties: On the , which cover , there are isomorphisms with a free module, and if we connect two such isomorphisms then the transition map is linear. These two properties give rise to what is called a locally free sheaf (the second condition is somehow hidden in the coherence. It will be explicit in the equivalent definition of a geometric vector bundle below).

We will now give the precise definition. For this we will work in the context of schemes. If you are not familiar with the theory of schemes, it is enough to think of as the spectrum of a ring or an open subset of it defined by an ideal (such schemes are called quasiaffine). Recall that consists of all prime ideals of together with the Zariski topology where a basis is given by

If , where is a field, then one should consider as the usual affine space , where the points correspond to the maximal ideals of the form (and all maximal ideals with residue class field are of this form). If , then one should think of as the closed subset of affine space consisting of the points such that .

For some also the word sheaf might be scary. As a first good approximation, one may think of a quasicoherent sheaf as an -module together with the family of localizations which are associated to the open subsets ().

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

The easist locally free sheaves are (), these are called free. The definition says exactly that locally a locally free sheaf is such a free sheaf. Over a local ring, any locally free sheaf is free, so there is not much to say. However, if we consider over a local ring the modules which are locally free outside the unique closed point of , i.e. on (which is called the punctured spectrum), then this is already a very important class of modules. Examples of this type will be the first syzygy module for an -primary ideal.

The following two theorems give equivalent characterizations of locally free sheaves on an affine scheme . Basically it says that the term locally can be understood in any meaningful sense.

Let denote a commutative noetherian ring and let denote a finitely generated -module. Let . Then the following conditions are equivalent.

- The localizations are free of rank for every prime ideal .
- The localizations are free of rank for every maximal ideal of .
- There exists elements which generate the unit ideal and such that the localizations are free of rank for every .
- The coherent sheaf on associated to is locally free.

Let denote a commutative noetherian ring and let denote a finitely generated -module. Let . Then the following conditions are equivalent.

- is locally free.
- is a projective module.
- is a (faithfully) flat module.

The following theorem provides many locally free sheaves. The syzygy sheaves discussed above are a special case of this construction, since they are the kernel of , which is surjective on .

Let denote a scheme and let and denote two locally free sheaves (of rank and ) together with a surjective sheaf homomorphism

** Then the kernel sheaf is also locally free
(of rank ). **

*Geometric vector bundles*

We develop an equivalent but more geometric notion for a locally free sheaf. Both concepts are equally important, and it is good to switch from one perspective to the other.

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 restrict always 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. 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 is independent of the chosen 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 vector space over .

This given we can say that a vector bundle is in particular a commutative group scheme (but one which is defined over an arbitrary base , not over the spectrum of a field), meaning that we have morphismus

fulfilling certain natural arrow-conditions expressing associativity, that is the neutral element and that gives the negative. This viewpoint will be later important when we have a look at the torsors of this group scheme.

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.

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 .

For a surjective morphism

on a scheme given by elements (the surjectivity means that these elements generate locally the unit ideal) we can realize the corresponding locally free kernel sheaf in the following natural way. We can directly look at the corresponding surjection of geometric vector bundles

and the kernel consists for every base point in the solution set

to this linear equation over the residue class field . So fiberwise this syzygy bundle is a very simple object, but of course the solution space varies with the basis. If is affine, then one can also describe the syzygy bundle as the spectrum of the -algebra

If the elements do not generate the unit ideal in , then the syzygy module yields only a vector bundle on the open subset . However, the algebra just mentioned,

always gives rise to a commutative group scheme over . Note that

The coadditon is given by

and the addition is given by

The zero element and the negatives are also defined in an obvious way. Also, the fibers of over a point is always a vector space over the residue class field. However, the dimension may vary. If is a point where all the functions vanish (and such points exist if these elements do not generate the unit ideal), then the equation which defines degenerates to and then the dimension of the fiber is instead of . This corresponds to the property that the linear equation degenerates and hence the dimension of the solution space goes up. In the next lecture we will study torsors of vector bundles and forcing algebras, which correspond to inhomogeneous linear equations varying with a basis.