Locally Free Sheaves and Vector Bundles

Posted on 2025-06-19 In Math

Outline

Section 2: Locally Free Sheaves and Vector Bundles

2 Locally Free Sheaves and Vector Bundles

The set T (M) of all homogeneous rst order operators on M has some nice structure.

Firstly it forms a vector space over R in an obvious way. In fact, if D

, D

∈ T (M) then

+ D

is dened by

+ D

)(φ) = D

φ + D

for all φ ∈ A(M ). It is obvious that it is also a derivation. Moreover if f ∈ A(M ) then

fD dened by

(fD)(φ) = fDφ

is also a derivation, thus making T (M ) an A(M )-module. Thirdly, if D

, D

are two

such operators, then the operator [D

, D

] dened by

, D

](φ) = D

(φ)) − D

(φ))

is also one such. For, if φ, ψ ∈ A(M), then we have

, D

](φ · ψ) = D

(φ · ψ)) − D

(φ · ψ))

= D

((D

φ)ψ + φ(D

ψ)) − D

((D

φ)ψ + φ(D

ψ))

= (D

φ))ψ + (D

φ)(D

ψ) + (D

φ)(D

ψ) + φ(D

ψ))

− (D

φ))ψ − (D

φ)(D

ψ) − (D

φ)(D

ψ) − φ(D

ψ))

= ([D

, D

]φ)ψ + φ([D

, D

]ψ).

We will refer to the map (D

, D

) 7→ [D

, D

] as the bracket operation. Finally, since any

D ∈ T (M ) may also be described as a sheaf homomorphism of A into itself satisfying

the Leibniz rule, it is clear that if V ⊂ U are open subsets of M, a restriction map

T (U) → T (V ) may also be dened, making the assignment to any U of T (U) a sheaf

as well. This sheaf will be denoted T .

2.1. Denition. A Lie algebra over a commutative ring k is a module V over k with

a k-bilinear operation

(X, Y ) 7→ [X, Y ]

which satises

i) [X, X] = 0;

ii) (Jacobi’s identity) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0,

for all X, Y, Z ∈ V . A homomorphism of a Lie algebra V into another Lie algebra W is

a k-linear map f : V → W which satises

f([X, Y ]) = [f (X), f(Y )]

Chapter II: Differential Operators

for all X, Y ∈ V .

2.2. Examples.

1) Consider the n

-dimensional vector space M(n, k) of (n, n) matrices with coe-

cients in a commutative ring k. Dene the bilinear map (X, Y ) 7→ XY −Y X to be

the bracket operation (using matrix multiplication). Then it is a Lie algebra over

k. More generally, any (associative) algebra can be regarded as a Lie algebra by

dening [x, y] = xy −yx where on the right side we use the algebra multiplication.

If V is a vector space of dimension n, the algebra End(V ) can be regarded as a

Lie algebra and this is the abstract version of the matrix Lie algebra.

2) We can use various subspaces of End(V ) or M(n) which are closed under the

bracket operation and obtain Lie algebras. For example, if b is a bilinear form on

V , we may consider {X ∈ End(V ) : b(Xv, w) + b(v, Xw) = 0 for every v, w ∈ V }.

It is easy to check that if X and Y satisfy this condition, XY − Y X also does.

3) Any vector space with the bracket operation dened by [v, w] = 0 gets a Lie

algebra structure. A Lie algebra in which all the bracket operations are 0 is called

an abelian Lie algebra.

4) Let A be any associative algebra over k. Then the space of k-derivations of A form

a Lie algebra over k under the operation [D

, D

] = D

− D

, for any two

derivations D

, D

The relevance of this denition here is of course that the bracket operation we dened

above on the space of homogeneous rst order operators, endows it with the structure

of a Lie algebra over R. Although T (M ) is a Lie algebra over R and is a module over

A(M), it is not a Lie algebra over A ( M ) since the bracket operation we dened above

is not A(M)-bilinear. In fact, we have

, fD

](φ) = D

(f(D

φ)) − (fD

)(D

(φ))

= D

f · D

φ + fD

φ) − fD

φ)

= (D

f · D

+ f[D

, D

])(φ).

It is obvious that the restriction maps T (U ) → T (V ) are all Lie algebra homomorphisms

so that the sheaf T is a sheaf of Lie algebras over R. Besides, we now have the following

situation. On the one hand, A and T are sheaves on M and on the other T (U ) is

an A(U)-module for every open subset U of M. The module structure is compatible

with the restriction maps in an obvious sense. We recall the denition of a sheaf of

A-modules, already indicated in [Ch. 1, 2.7]

2.7.

Section 2: Locally Free Sheaves and Vector Bundles

2.3. Denition.

i) Let A be a sheaf of rings over a topological space X. Let M be a sheaf of abelian

groups over X with a structure of A(U)-module on M(U) for every open subset

U of X. Then we say that M is a sheaf of A-modules or that it is an A-module, if

res

(am) = (res

a)(res

(m))

for all a ∈ A(U) and m ∈ M(U ) and open sets V ⊂ U .

ii) An A-homomorphism of an A-module M

into another A-module M

is a ho-

momorphism f of sheaves of abelian groups such that the homomorphisms f (U) :

(U) → M

(U) are A(U )-linear, for all open sets U in X.

2.4. Examples.

1) With this denition, we see that if (M, A) is a dierential manifold, then T is an

A-module, besides being a sheaf of Lie algebras over R. This sheaf will be called

the tangent sheaf.

2) The direct sum A

of A with itself r times (where A is a sheaf of rings) is obviously

an A-module.

3) If (M, A) is a dierential manifold and Z ⊂ M is a closed set, one may consider for

each open subset U of M , the set of all elements of A(U ) which vanish on Z ∩U.

This gives a sheaf I

which is clearly an A-module. This is called the ideal sheaf

of Z in M .

The local structure of the sheaf T as an A-module is quite simple. We have seen

that if (U, x) is a coordinate system, the sheaf T |U is actually isomorphic to A

, the

map (a

) 7→

∑

∂

∂x

providing such an isomorphism.

2.5. Denition. Let A be a sheaf of rings on a topological space X. An A-module

M is said to be locally free of rank r if every x ∈ X has a neighbourhood U such that

the restriction of M to U is isomorphic to A

|U as an A|U -module.

With this denition, our observation above implies that the tangent sheaf of a dif-

ferential manifold is locally free as an A-module, where A is the sheaf of dierentiable

functions.

2.6. Example. Let N be a closed submanifold of a dierential manifold M . We

dened in 2.4, Example 3) the sheaf of ideals I by the prescription I(U) = {f ∈ A(U ) :

f(x) = 0 for all x ∈ U ∩ N }. If N has dimension n −1, then this sheaf is a locally free

A-module of rank 1. In order to see this, rst observe that this being a local statement,

we may assume that M is the unit ball in R

and that N is the closed submanifold

Chapter II: Differential Operators

given by x

= 0. If f is a function vanishing on N , then it can be written as x

· g. In

fact, we have

f(x

, . . . , x

) =

∫

∂

∂t

f(x

, . . . , tx

)dt = x

∫

∂f

∂x

, . . . , tx

)dt.

In other words, the ideal of functions vanishing on N is a principal ideal generated by

Any function f which generates the ideal of functions vanishing on N, can be written

as x

· g with g nonvanishing. This implies that at any point of N , the function

∂f

∂x

g +x

∂g

∂x

does not vanish at any point of N. In other words, a function f is a generator

of the ideal at 0 if and only if it vanishes on

and at least one of the partial derivatives

of f is nonzero at 0.

2.7. Exercise. Show that the R-linear map of the maximal ideal M of functions

vanishing at 0 into R

taking f to

((

∂f

∂x

)

, . . . ,

(

∂f

∂x

)

induces an isomorphism of

M/M

onto R

. Conclude that if r > 1, then the ideal sheaf M of 0 is not locally free.

2.8. Denition. Let A be a constant sheaf of rings. Then any locally free sheaf of

A-modules is called a local system.

Let (M, A) be a dierential manifold. Since sections of A are simply dierentiable

functions on U , it is natural to call sections of the A-module A

⊕

A, systems of

functions or vector-valued functions. In the general case of a locally free sheaf E, let

us discuss whether we can think of sections as some sort of functions. Locally it is

indeed possible since we have assumed that E is locally isomorphic to A

. If x ∈ M ,

and M

is the ideal of A

consisting of germs of functions at x which vanish at x,

one may consider for any f ∈ A

its image in A

. The natural evaluation map

→ R taking any f to f (x), gives an isomorphism of A

with R. Therefore one

may think of the evaluation map as taking the image of f in A

. Guided by this

fact, we may take the R-vector space E

to be the space in which the sections

of E take values at the point x ∈ M . The (important) dierence between the case of

A and the general case of a locally free sheaf, is that the vector space associated to

x ∈ M , namely E

= E

depends on the point x ∈ M . In other words, there is

no natural isomorphism of these vector spaces at two dierent points. Consider the set

union E of all these sets, namely

∪

x∈M

. This comes with a natural map π into M,

with π

−1

(x) = E

. For any x ∈ M there exists an open neighbourhood U such that

E|U is isomorphic to A

, so that π

−1

(U) may be identied with U × R

and the map

−1

(U) → U with the projection U × R

→ U. In particular, the set π

−1

(U) can be

provided with the dierential structure of the product. This is obviously independent

of the isomorphism E|U → A

chosen and hence may be patched together to yield a

dierential structure on E. It is easy to see that the topology we have introduced is

Section 2: Locally Free Sheaves and Vector Bundles

Hausdor and that it admits a countable base for open sets. Now we have the following

structures on E.

a) E is a dierential manifold.

b) π : E → M is a dierentiable map.

c) For any x ∈ M , there exist an open neighbourhood U of x and a dieomorphism

−1

(U) → U × R

such that π becomes the projection U ×R

→ U following this

isomorphism.

d) There is a structure of a vector space on each bre π

−1

(x) for every x ∈ M, which

is compatible with the isomorphism above. This means that if we identify π

−1

(x)

with R

using the local isomorphism in c) above, then the identication is a linear

isomorphism.

2.9. Denition. A dierentiable map π : E → M satisfying a) – d) above, is called

a dierentiable vector bundle of rank r. A homomorphism of a vector bundle E into F

is a dierentiable map E → F which makes the diagram

E F

commutative such that the induced maps on bres are all linear.

The isomorphism of the type described in c) above is referred to as a local triviali-

sation of the bundle E.

We have associated, to any locally free sheaf E of A-modules, a dierentiable vector

bundle E → M. Conversely, if π : E → M is a dierentiable vector bundle, then the

sheaf of dierentiable sections of π is a locally free sheaf of A-modules. Moreover, if F

is another locally free sheaf of A-modules with associated vector bundle F , then there

is a natural bijection between the set of A-linear sheaf homomorphisms E → F and the

set of vector bundle homomorphisms E → F . Thus one may use the notions of ‘locally

free sheaf’ and ‘dierentiable vector bundle’ interchangeably.

2.10. Remark. One has however to guard against the following possible confusion.

A subbundle of E is a vector bundle F together with a homomorphism F → E which is

injective on all bres. It gives rise to an injective A-homomorphism of F into E, namely

an A-subsheaf of E. Conversely, the inclusion of an A-module F in E gives rise to a

homomorphism F into E, but the latter need not be injective on all bres.

2.11. Exercise. Consider the inclusion of the ideal sheaf of any point in R in the

sheaf A. Show that the corresponding homomorphism at the vector bundle level is not

injective on all bres.

Chapter II: Differential Operators

If φ : M → M

′

is a dierentiable map and E is a dierentiable vector bundle on

′

, then one can dene a dierentiable vector bundle φ

∗

E called the pull-back of E as

follows. Take the subspace of M × E dened by

∗

E = {(m, x) ∈ M × E : φ(m) = π(x)}

It is obvious that it is Hausdor and admits a countable base for open sets. This

subspace comes with two natural maps, namely, π

′

: φ

∗

E → M given by π

′

(m, x) = m

and ˜φ : φ

∗

E → E given by ˜φ(m, x) = x. If U is an open set over which E is trivial and

V = φ

−1

(U), then π

′−1

(V ) can be identied with V × R

so that φ

∗

E is a dierential

manifold, with respect to which π

′

is dierentiable. Clearly the bre π

′−1

(m) over

m ∈ M can be identied with the bre π

−1

(φ(m)) and hence has a natural vector space

structure. It is now easy to verify that π

′

: φ

∗

E → M satises the conditions a) – d)

and hence φ

∗

E is a vector bundle.

2.12. Denition. The vector bundle φ

∗

E dened above is called the pull-back of

the vector bundle E by the map φ : M → M

′

. For every dierentiable section s of E

one can dene a section of φ

∗

E called the pull-back of s in such a way that

˜φ(φ

∗

s(m)) = s(φ(m))

for all m ∈ M .

Suppose E is a locally free sheaf of A-modules on M

′

and φ is a dierentiable map

M → M

′

. Then the inverse image φ

−1

′

as dened in Chapter 1, 2.3 is a sheaf of

algebras over M. The inverse image φ

−1

(E) is not a sheaf of A

-modules but only a

locally free sheaf of φ

−1

′

)-modules. The companion map of φ, namely the map

φ : φ

−1

′

) → A

may be used to dene φ

∗

E as φ

−1

E ⊗

−1

′

)

, where A

considered as a sheaf of φ

−1

′

)-modules via φ. With this denition it is easy to see

that the vector bundle associated to φ

∗

E can be canonically identied with the pull-back

∗

E of the vector bundle E associated to E.

2.13. Remarks.

1) All the constructions which we have made above are also valid for locally free

sheaves of A

⊗ C-modules. The corresponding vector bundles are bundles of

vector spaces over C. If E is a complex vector bundle, one can dene E by setting

E = E as a dierential manifold but changing the vector space structure on the

bres of E by redening multiplication by i =

√

−1 as multiplication by −i in E.

Given any real vector bundle E one can associate to it a complex vector bundle E

by tensoring with C. Of course in such a case, we have a canonical isomorphism

of E

with E

2) We refer to complex vector bundles of rank 1 as line bundles. Real vector bundles

Section 2: Locally Free Sheaves and Vector Bundles

of rank 1 are not that interesting. In fact, if L is one such, then consider the

relation given by declaring two points v, v

′

of L to be equivalent if there exists

a positive real number a such that v

′

= av. Then the quotient space maps onto

M and has precisely two points on each bre. In view of the local triviality of

the line bundle, the quotient is easily seen to be a two-sheeted (étale) covering

space M

′

of M . If M is simply connected, this covering consists therefore of two

copies of M . Choosing one of them is equivalent to choosing for all m ∈ M , one

of the components of L

\ {0}, where L

is the bre of L at m. We may call

elements of that component positive vectors. Local trivialisation of L is equivalent

to the data consisting of an open covering {U

} and sections s

on each U

which

are everywhere nonzero. We may in the above situation, actually choose positive

sections s

, i.e. sections whose values are in the chosen component in the bre.

We may then take a partition of unity {φ

} for the covering and get a global

everywhere nonzero section

∑

of L. This shows that L is globally trivial.

This argument actually shows that even if M were not simply connected, we can

pull back L to the two-sheeted covering M

′

and trivialise it on M

′

3) If M is a closed submanifold of R

of dimension n − 1, then we have remarked

already in (2.6) that its ideal sheaf is a locally free A-module of rank 1. By 2)

this sheaf is globally trivial. Therefore there is a section which generates the local

ideal everywhere, that is to say, there is a function f which vanishes precisely on

M and generates the ideal sheaf locally at all points.

Let us get back to our sheaf T , the tangent sheaf of M. Applying the above con-

siderations to it, we get a vector bundle, which is called the tangent bundle T . For any

x ∈ M , the bre over x is called the tangent space at x. Elements of this space are

called tangent vectors at x. A dierentiable section of this vector bundle may therefore

be called a vector eld and thus the concept of a homogeneous dierential operator of

rst order and that of a vector eld are essentially equivalent.

The tangent space at a point x ∈ M is thus the quotient space T

. If X is a

germ of a vector eld at x then the map f 7→ Xf induces a derivation of the R-algebra

. The map f 7→ (Xf )(x) gives rise to a map t : A

→ R which is a derivation in the

sense that

t(fg) = (tf )g(x) + f (x)t(g)

for all f, g ∈ A

. By this correspondence, one easily veries that the tangent space at x

can be identied with the set of linear maps A

→ R satisfying the above condition.

In the case when M = R

, or more abstractly a vector space V of dimension n, the

tangent bundle is trivial. The tangent space at any point v ∈ V can be identied with

the vector space itself. In fact, we associate to any x ∈ V the derivation ∂

: A

→ R

given by f(x) 7→ lim

t→0

f(x+tv)−f (x)

. If N is a submanifold of V , then the tangent

Chapter II: Differential Operators

bundle of N is a subbundle of the trivial bundle N ×V . Thus the tangent space at any

point P ∈ N is a subspace of V . We visualise this geometrically as the coset space of

this subspace in V which contains P . In other words, the geometric tangent space is

then the space parallel to the abstract tangent space, passing through P .

2.14. Exercise. If a submanifold N of R

is given by f = 0 where f is a dierentiable

function with the property that at least one of the partial derivatives of f is nonzero at

every point of N , write down the equation of the tangent space at a point n ∈ N.

2.15. Dierential of a map. If φ : M → M

′

is a dierentiable map of a manifold

M into M

′

and t is a tangent vector at m ∈ M, then one can dene a tangent vector

φ(t) at m

′

= φ(m) , by setting φ(t)(f) = t(f ◦ φ) for all f ∈ A

′

. This denes a linear

map T

(M) → T

′

) called the dierential of φ. The tangent space at m is to be

thought of as the linear approximation of M near m and the dierential of φ as the

linear approximation of the map φ. Globally speaking, the dierential gives a vector

bundle homomorphism of T

into φ

∗

′

), or what is the same, a sheaf homomorphism

of T

into φ

∗

′

). This homomorphism is usually denoted by dφ. Occasionally, we

may write φ for this dierential as well.

If (U, x) (resp. (V, y)) is a coordinate system in M (resp. M

′

), such that φ(x) = y

and φ(U) ⊂ V , then φ is given by functions φ

, . . . , x

). Then its dierential takes

∂

∂x

to the vector at φ(x) which takes the coordinate functions y

∂

∂x

(φ

(x)). Hence

we deduce that the dierential of φ maps

∂

∂x

∑

∂

∂x

(φ

(x))

∂

∂y

. In other words, the

matrix of the linear map T

→ T

φ(x)

with respect to the standard bases, is given by

∂φ

∂x

(φ

(x)). This matrix is called the Jacobian of the map φ.

If f is a dierentiable function, namely a dierentiable map M → R, then df is thus

a homomorphism T

→ f

∗

). But T

is a trivial bundle since the vector eld

forms a basis for the tangent spaces at all points of R. Hence df may be regarded as a

homomorphism of T

into the trivial bundle of rank 1, or what is the same, a section

of the dual vector bundle. This may simply be called the dierential of f.

Section 2: Locally Free Sheaves and Vector Bundles

2.16. Normal bundle. If f : N → M is a dierentiable map such that the dierential

of f is injective at all points of N , then the dierential of f, namely the homomorphism

→ f

∗

), is a subbundle inclusion. The cokernel of this is called the normal bundle

of N in M . Thus we have the exact sequence of vector bundles:

0 → T

→ f

∗

) → Nor(N, M ) → 0

Under this assumption, the implicit function theorem assures us of the existence

of a local coordinate system (U, x) at a point n ∈ N and a local coordinate system

(V, y) at f(n) such that f|U is injective and f (U ) ⊂ V is given by the vanishing of

some of the coordinates, say y

, . . . , y

. The tangent space to N is freely generated by

∂

∂y

r+1

, . . . ,

∂

∂y

If in addition f is injective, it follows that N is a manifold immersed in M . In that

case, the notion of normal bundle is more geometric. Suppose now that N is a closed

submanifold of R

. Then the tangent space at P ∈ N is a subspace of R

, and the

normal space is a quotient. But using the Euclidean metric on R

, we may identify this

quotient with a subspace as well. We may visualise it as the coset of this latter space,

passing through P .

2.17. Exercises.

1) Let N be a submanifold of R

given by f = 0 as in 2.14. Write down the equation

of the normal space at any point n ∈ N .

2) If M is a closed submanifold of R

of dimension n−1, show that its normal bundle

is trivial.

On the other hand, the dierential of f may even be an isomorphism at all points,

without the map f being injective.

Chapter II: Differential Operators

2.18. Denition. If the dierential of f : N → M is an isomorphism at all points of

N, we call it an étale map.

2.19. Example. The map t 7→ exp(it) of R → S

is an étale map.

If an étale map is also injective, it follows from the denition that it is an inclusion

of an open submanifold.