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A general theory of fibre spaces
with structure sheaf
Alexander Grothendieck
Introduction. When one tries to state in a general algebraic formalism the various
notions of fibre space : general fibre space (without structure group, and maybe not
even locally trivial) ; or fibre bundle with topological structure group G as expounded
in the book of Steenrod (The Topology of Fibre Bundles, Princeton University Press) ;
or the “differentiable” and “analytic” (real or complex) variants of these notions ; or the
notions of algebraic fibre spaces (over an abstract field k ), one is led in a natural way
to the notion of fibre space with a structure sheaf G. This point of view is also
suggested a priori by the possibility, now classical, to interpret the (for instance
“topological”) classes of fibre bundles on a space X , with abelian structure group G ,
as the elements of the first cohomology group of X with coefficients in the sheaf G of
germs of continuous maps of X into G ; the word “continuous” being replaced by
“analytic” respectively “regular” if G is supposed an analytic respectively an algebraic
group (the space X being of course accordingly an analytic or algebraic variety). The
use of cohomological methods in this connection have proved quite useful, and it has
become natural, at least as a matter of notation, even when G is not abelian, to
denote by H
1 ( X, G ) the set of classes of fibre spaces on X with structure sheaf G , G
being as above a sheaf of germs of maps (continuous, or differentiable, or analytic, or
algebraic as the case may be) of X into G. Here we develop systematically the notion
of fibre space with structure sheaf G , where G is any sheaf of (not necessarily
abelian) groups, and of the first cohomology set H
1 ( X, G ) of X with coefficients in G.
The first four chapters contain merely the first definitions concerning general fibre
spaces, sheaves, fibre spaces with composition law (including the sheaves of groups)
and fibre spaces with structure sheaf. The functor aspect of the notions dealt with has
been stres sed throughout, and as it now appears should have been stressed even
more. As the proofs of most of the facts stated reduce of course to straightforward
verifications, they are only sketched or even omitted, the important point being merely
a consistent order in the statement of the main facts. In the last chapter, we define the
cohomology set H
1 ( X, G ) of X with coefficients in the sheaf of groups G , so that the
expected classi fication theorem for fibre spaces with structure sheaf G is valid. We
then proceed to a careful study of the exact cohomology sequence associated with an
exact sequence of sheaves e −→ F −→ G −→ H −→ e. This is the main part, and in
fact the origin, of this paper. Here G is any sheaf of groups, F a subsheaf of groups, H
= G/F , and according to various supplementary hypotheses on F (such as F normal,
or F normal abelian, or
Cours donné à l’Université du Kansas, à Lawrence, aux Etats-Unis (NSF-G 1126, Projet de
recherche sur la géométrie des espaces de fonctions, Rapport n
o 4, première édition août 1955, seconde
édition Mai 1958).
F in the center) we get an exact cohomology sequence going from H
0 ( X, F ) (the group
of sections of F ) to H
1 ( X, G ) respectively H
1 ( X, H ) respectively H
2 ( X, H ), with more or
less additional algebraic structures involved.
The formalism thus developed is quite suggestive, and as it seems useful, in
particular in dealing with the problem of classification of fibre bundles with a structure
group G in which we consider a sub-group F , or the problem of comparing say the
topological and analytic classification for a given analytic structure group G. However,
in order to keep this exposition in reasonable bounds, no examples have been given.
Some com plementary facts, examples, and applications for the notions developed
will be given in the future. This report has been written mainly in order to serve the
author for future reference ; it is hoped that it may serve the same purpose, or as an
introduction to the subject, to somebody else.
Of course, as this report consists in a fortunately straightforward adaptation of quite
well known notions, no real difficulties had to be overcome and there is no claim for
originality whatsoever. Besides, at the moment to give this report for mimeography, I
hear that results analogous to those of chapter 5 were known for some years to Mr.
Frenkel, who did not publish them till now. The author only hopes that this report is
more pleasant to read than it was to write, and is convinced that anyhow an
exposition of this sort had to be written.
Remark (added for the second edition). It has appeared that the formalism developed
in this report, and specifically the results of Chapter V, are valid (and useful) also in
other situations than just for sheaves on a given space X. A generalization for
instance is obtained by supposing that a fixed group Π is given acting on X as a group
of homeomorphisms, and that we restrict our attention to the category of fibre spaces
over X (and especially sheaves) on which Π operates in a manner compatible with its
operations on the base X (See for instance A. Grothendieck, Sur le mémoire de Weil ;
Généralisation des fonctions abéliennes, Séminaire Bourbaki Décembre 1956). When
X is reduced to a point, one gets (instead of sheaves) sets, groups, homogeneous
spaces etc, admitting a fixed group Π of operators, which leads to the (commutative
and non commutative) cohomology theory of the group Π. One can also replace Π by
a fixed Lie group (operating on differentiable varieties, on Lie groups, and
homogeneous Lie spaces). Or X, Π are replaced by a fixed ground field k , and one
considers algebraic spaces, algebraic groups, homogeneous spaces defined over k ,
which leads to a kind of cohomology theory of k. All this suggests that there should
exist a comprehensive theory of non-commutative cohomology in suitable categories,
an exposition of which is still lacking. (For the “commutative” theory of cohomology,
see A. Grothendieck, sur quelques points d’Algèbre Homologique, Tohoku Math.
Journal, 1958).
onto-homeomorphisms.
1.2 Inverse image of a fibre space, inverse homomorphisms.
Let ( X, E, p ) be a fibre space over the space X , and let f be a continuous map of a
space X
0 into X. Then the inverse image of the fibre space E by f is a fibre space E
0
over X
0
. E
0 is defined as the subspace of X
0 ×E of points ( x
0 , y ) such that fx
0 = py , the
projection p
0 of E
0 into the base X
0 being given by p
0 ( x
0 , y ) = x
0
. The map g ( x
0 , y ) = y
of E
0 into E is then an f -homomorphism, inducing for each x
0 ∈ X
0 a homeomorphism
of the fibre of E
0 over x
0 onto the fibre of E over f
0 x
0 .
Suppose now, moreover, given a continuous map f
0 : X
00 −→ X
0 of a space X
00 into X
0 .
Then there is a canonical isomorphism of the fibre space E
00 over X
00 , inverse image
of the fibre space E by ff
0 , and the inverse image of the fibre space E
0 (considered
above) by f
0 (transitivity of inverse images). If ( x
00 , y ) ∈ E
00 ( x
00 ∈ X
00 , y ∈ E, ff
0 x
00
py ), it is mapped by this isomorphism into ( x
00 , ( f
0 x
00 , y )).
Let Y be a subspace of the base X of a fibre space E ; consider the injection f of Y into
X ; the inverse image E
0 of E by f is called fibre-space induced by E on Y , or the
restriction of E to Y , and is denoted by E|Y. This is canonically homeomorphic to a
subspace of E , namely the set of elements mapped by p into Y ; the projection of E|Y
into Y is induced by p. By what has been said above, if Z is a subspace of Y , the
restriction of E|Y to Z is the restriction E|Z of E to Z.
Again let ( X, E, p ) and ( X
0 , E
0 , p
0 ) be two fibre spaces, f a continuous map X −→ X
0 .
An inverse homomorphism associated with f is an X -homomorphism g of the fibre
space E 0 into E , where E 0 denotes the inverse image of the fibre space E
0 by f. That
means that g is a continuous map, of the subspace E 0 of X × E
0 of pairs ( x, y
0 ) such
that fx = p
0 y
0 , into E , mapping for any x ∈ X the fibre of x into E 0 (homeomorphic to
the fibre of fx in E
0 !) into the fibre p
− 1 ( x ) of x in E. For instance, if E is itself the inverse
image of E
0 by f , then there is a canonical inverse homomorphism of E
0 into E
associated with f : the identity! (Though somewhat trivial, this is the most important
case of inverse homomorphisms.)
1.3 Subspace, quotient, product.
Let ( X, E, p ) be a fibre space, E
0 any subspace of E , then the restriction p
0 of p to E
0 ,
defines E
0 as a fibre space with the same basis X , called a sub-fibre-space of E. So
the sub-fibre-spaces of E are in one to one correspondence with the subsets of E ; in
particular, for them the notions of union, intersection etc. are defined. (Of course, in
most cases we are only interested in fibre spaces the projection of which is onto ; this
imposes then a condition on the subspaces of E considered, which may be fulfilled for
two subspaces and not for the intersection.)
Let now R be an equivalence relation in E compatible with the map p , i.e. such that
two elements of E congruent mod R have the same image under p. Then p defines a
continuous map p
0 of the quotient space E
0 = E/R into X , which turns E
0 into a fibre
space with base X , called a quotient fibre space of E. So the latter are in one-to-one
correspondence with the equivalence relations in E compatible with p. A quotient fibre
space of a quotient fibre space of E is a quotient fibre space.
Let ( X, E, p ) and ( X
0 , E
0 , p
0 ) be two fibre spaces, then ( p, p
0 ) defines a continuous
map of E × E
0 into X × X
0 , so that E × E
0 appears as a fibre space over X × X
0 , called
the
product of the fibre spaces E, E
0 .
The fibre of ( x, x
0 ) in E × E
0 is the product of the fibres of x in E , respectively x
0 in E
0 .
Suppose now X = X
0 , and consider the inverse image of E ×E
0 under the diagonal
map X −→ X × X , we get a fibre space over X , called the fibre product of the fibre
spaces
E, E
0 over X , denoted by E × ( X )
E
0
. The fibre of x in this fibre-product is the
product
of the fibres of x in E respectively E
0
. Of course, product of an arbitrary family of fibre
spaces can be considered, and the usual formal properties hold.
1.4 Trivial and locally trivial fibre spaces.
Let X and F be two spaces, E the product space, the projection of the product on X
defines E as a fibre space over X , called the trivial fibre space over X with fibre F.
All fibres are canonically homeomorphic with F. Let us determine the homomorphisms
of a trivial fibre space E = X × F into another E
0 = X × F
0
. More generally, we will only
assume that the projection of X × F onto X is the natural one and continuous for the
given topology of X × F , which induces on the fibres the given topology (but the
topology of X ×F may not be the product topology, for instance : X and F are algebraic
varieties with the Zariski topology) ; same hypothesis on X ×F
0
. Then a
homomorphism u of E into E
0 , inducing for each x ∈ X a continuous map of the fibre
of E over x into the fibre of E
0 over x , defines a function x −→ f ( x ) of X into the set of
all continuous maps of F into F
0 , and of course the homomorphism is well determined
by this map by the formula
triple ( i, j, k ) of indices such that Uijk = Ui ∩ Uj ∩ Uk 6 = ∅, the relation
(1.5.2.) fik = fijfjk
(where, in order to abbreviate notations, we wrote simply fik instead of : the isomor
phism of Ek|Uijk onto Ei|Uijk induced by fik and likewise for fij and fjk ). Supposing this
condition satisfied, let E be the quotient space of E by the preceding equivalence
relation. The projections pi of Ei into Ui define a continuous map of the topological sum
E into X , and this map is compatible with the equivalence relation in E, so that there is
a continuous map p of E into X (which is onto if the pi ’s are all onto).
Definition 1.5.1. The fibre space over X just constructed is called the fibre space
defined by the “coordinate transformations” ( fij ) between the fibre spaces Ei.
The identity map of Ei into E defines a map S i , of Ei into E , which by virtue of (1.5.1.) is
a one to one Ui -homomorphism of Ei onto E|Ui. The topology of E (by a well known
transitivity property for topologies defined as the finest which ...) is the finest topology
on E for which the maps S i are continuous. Moreover, it is easy to show that in case
the interiors of the Ui ’s already cover X , the maps S i are homeomorphisms into. Hen
ceforth, for simplicity we will only work with open coverings of X , so that the preceding
properties are automatically satisfied. Then S i can be considered as a Ui -isomorphism
of Ei onto E|Ui. Clearly
(1.5.3.) fij = S
− 1
i S j
(where again, in order to abbreviate, we wrote S i instead of the restriction of S i to Ei|Uij
, S j instead of the restriction of S j to Ej |Uij ). Conversely, let E be a fibre space over X ,
and suppose that for each i , there exists a Ui -isomorphism S i of Ei onto E|Ui , then
(1.5.3.) defines, for each pair ( i, j ) such that Ui ∩Uj = Uij 6 = ∅, a Uij -isomorphism of Ej
|Uij onto Ei|Uij , and the system ( fij ) satisfies obviously (1.5.2.). Therefore we can
consider the fibre space E
0 defined by the coordinate transformations fij. Then it is ob
vious that the map of E into E defined by the maps S i is compatible with the
equivalence relation in E, therefore defines a continuous map f of E
0 into E which is of
course an X -homomorphism. Let S
0 i be the natural isomorphism of^ Ei onto^ E
0 |Ui
defined above ; it is checked at once that the map of E
0 |Ui into E|Ui induced by f is
S i S
0 i
− 1 , hence an isomorphism onto. It follows that f itself is an isomorphism of E
0 onto
E , by virtue of the following easy lemma (proof left to the reader) :
Lemma 1. Let E, E
0 be two fibre spaces over X , and f an X -homomorphism of E into
E
0 , such that for any x ∈ X , exists a neighborhood U of x such that f induces an
isomorphism of E|U onto (respectively, into) E
0 |U. Then f is an X -isomorphism of E
onto (respectively, into) E
0 .
What precedes shows the truth of :
Proposition 1.5.1. The open covering ( Ui ) and the fibre spaces Ei over Ui being given,
the fibre spaces over X which can be obtained by means of suitable coordinate trans
formations ( fij ) are exactly those, up to isomorphism, for which E|Ui is isomorphic to Ei
for any i.
Consider now two systems of coordinate transformations ( fij ) , ( f
0 ij ) corresponding to
the same covering ( Ui ), and to two systems ( Ei ), ( E
0 i ) of fibre spaces over the^ Ui ’s. Let
E be the fibre space defined by ( fij ) and E
0 the fibre space defined by ( f
0 ij ); we will
determine all homomorphisms of E into E
0
. If f is such a homomorphism, then for
each i , fi = S
0 i
− 1 f S i (where f stands for the restriction of f to E|Ui ) is a homomorphism of
Ei into E
0 i , and the system ( fi ) satisfies clearly, for each pair ( i, j ) such that^ Uij 6 =^ ∅^ :
(1.5.4) fifij = f
0 ijfj
(where we write simply fi instead of the restriction of fi to Ei|Uij , and likewise for fj ). The
homomorphism f is moreover fully determined by the system ( fi ) since fi deter mines
the restriction of f to E|Ui ; and moreover the system ( fi ) subject to (1.5.4) can be
chosen otherwise arbitrarily, for this relation expresses exactly that the map of the
topological sum E of the Ei ’s into the topological sum E
0 of the E
0 i
’s transforms equi
valent points into equivalent points, and therefore defines an X -homomorphism f of E
into E
0 ; and it is clear that the system ( fi ) is nothing else but the one which is defined
as above in terms of the homomorphism f. Of course, in view of lemma 1, in order that
f be an isomorphism onto, (respectively, into) it is necessary and sufficient that each fi
be an isomorphism of Ei onto (respectively, into) E
0 i. Thus we get :
Proposition 1.5.2. Given two fibre spaces over X , E and E
0 , defined by coordinate
transformations ( fij ) respectively ( f
0 ij ) relative to the same open covering ( Ui ), the
X -homomorphisms f of E into E
0 are in one to one correspondence with systems ( fi ) of
Ui -homomorphisms Ei −→ E
0 i satisfying (1.5.4.).^ f^ is an onto-isomorphism if and only if
the f
0 i ’s are, i.e.^ E
0 is isomorphic to E if and only if we can find onto-isomorphisms fi : Ei
−→ E
0 i such that, for any pair ( i, j ) of indices satisfying^ Uij 6 =^ ∅, we have
(1.5.5.) f
0 ij =^ fifijf
− 1
j
In case T is a trivial fibre space, T = X ×F , we have Ei = Ui ×F , and Ei|Uij = Uij ×F. Thus
fij is an automorphism of the trivial fibre space Uij × F , and therefore, in view of
proposition 1.4.1. given by a map x −→ fij ( x ) of Uij into the group of homeomorphisms
of F onto itself. The equations (1.5.2.) expressing that ( fij ) is a system of coordinate
transformations then translate into
(1.6.1.) fik ( x ) = fij ( x ) fjk ( x ) for x ∈ Uijk
Moreover, it must not be forgotten that x −→ fij ( x ) is submitted to the continuity
condition of proposition 1.4.1. Such a system then defines in a natural way a fibre
space E over X , and by what has been said it follows that this fibre bundle is locally
isomorphic to X × F , i.e. locally trivial with fibre F , and that (for suitable choice of the
covering and the coordinate transformations), we get thus, up to isomorphism, all
locally trivial fibre spaces over X with fibre F.
Let in the same way T
0 = X × F
0 , and consider for the same covering ( Ui ) a system ( fij )
and a system ( f
0 ij ) of coordinate transformations, the first relative to the fibre^ F^ and
the second to the fibre F
0
. Let E and E
0 be the corresponding fibre spaces over X. The
homomorphisms of E into E
0 , by proposition 1.5.2., correspond to homomorphisms fi
of Ei = Ui × F into E
0 i =^ Ui × F
0 , satisfying conditions (1.5.4). Now, (proposition 1.4.1.)
such a homomorphism fi is determined by a map x −→ fi ( x ) of Ui into the set of
continuous maps of F into F
0 by fi ( x, y ) = ( x, fi ( x ) .y ), subject to the only requirement
that fi ( x ) .y is continuous with respect to the pair ( x, y ) ∈ Ui × F. Then the equation
(1.5.4.) translates into
(1.6.2.) fi ( x ) fij ( x ) = f
0 ij ( x ) fj ( x ) ( x^ ∈^ Uij )
Thus are determined the homomorphisms of E into E
0
. In particular, the isomorphisms
of E onto E
0 are obtained by systems ( fi ) such that fi ( x ) be a homeomorphism of F
onto F
0 for any x ∈ Ui , and that x −→ f
− 1
i ( x ) satisfies the same continuity requirement
as x −→ fi ( x ). The compatibility condition (1.6.2.) can then be written
(1.6.3.) f
0 ij ( x ) =^ fi ( x ) fij ( x ) fj ( x )
− 1 ( x ∈ Uij )
1.7 Sections of fibre spaces.
Definition 1.7.1. Let ( X, E, p ) be a fibre space ; a section of this fibre space (or, by
pleo nasm, a section of E over X ) is a map x of X into E such that ps is the identity
map of X. The set of continuous sections of E is noted H
0 ( X, E ).
It amounts to the same to say that s is a function the value of which at each x ∈ X is
in the fibre of x in E (which depends on x !). The existence of a section implies of
course that p is onto, and conversely if we do not require continuity. However, we are
primarily interested in continuous sections. A section of E over a subset Y of X is by
definition a section of E|Y. If Y is open, we write H
0 ( Y, E ) for the set H
0 ( Y, E|Y ) of all
continuous sections of E over Y.
H
0 ( X, E ) as a functor. Let E, E
0 be two fibre spaces over X , f an X -homomorphism of E
into E
0
. For any section s of E , the composed map fs is a section of E
0 , continuous if s
is continuous. We get thus a map, noted f , of H
0 ( X, E ) into H
0 ( X, E
0 ). The usual
functor properties are satisfied :
a. If the two fibre spaces are identical and f is the identity, then so is f
b. if f is an X -homomorphism of E into E
0 and f
0 an X -homomorphism of E
0 into E
00
( E, E
0 , E
00 fibre spaces over X ) then ( f
0 f ) = f
0 f.
Let ( X, E, p ) be a fibre space, f a continuous map of a space X
0 into X , and E
0 the
inverse image of E under f. Let s be a section of E
0 consider the map s
0 of X
0 into E
0
given by s
0 x
0 = ( x
0 , sfx
0 ) (the second member belongs to E
0 , since fx
0 = psfx
0 because
px = identity), this is a section of E
0 , continuous if s is continuous. Thus we get a
canonical map of H
0 ( X, E ) into H
0 ( X
0 , E
0 ) ( E
0 being the inverse image of E by f ). In
case X
0 ⊂ X and f is the inclusion map, therefore E
0 = E|X
0 , then the preceding map
is nothing but the restriction map (of H
0 ( X, E ) into H
0 ( X
0 , E ) if X
0 open). We leave to
the reader statement and proof of an evident property of transitivity for the canonical
maps just considered.
The two sorts of homomorphisms for sets of continuous sections are compatible in
the following sense. Let S be a fixed continuous map of a space X
0 into X , then to any
fibre space E over X corresponds its inverse image E
0 under S, which is a fibre space
over X
0 ; moreover, given an X -homomorphism f : E −→ F , it defines in a natural way
an X
0 -homomorphism f
0 of E
0 into F
0
. (We could go further and state that, for fixed S,
E
0 is a “functor” of E by means of the preceding definitions.)
Then the following diagram
f ∗
H
0 ( X, E ) H
0 ( X, F )
relative to an open covering ( Ui ) of X and fibre spaces Ei over Ui. Then there is a
canonical one to one correspondence between sections of E and systems ( si ) of
sections of Ei over Ui , i satisfying conditions (1.7.3.). Continuous sections correspond
to systems of continuous sections.
Let again, as in section 1.5, be given two systems ( Ei ) and ( E
0 i ) of fibre spaces over
the Ui ’s and two corresponding systems of coordinate transformations ( fij ) and ( f
0 ij ) let
E and E
0 be the corresponding fibre spaces, and f an X -homomorphism of E into E
0 ,
defined by virtue of proposition 1.5.2., by a system ( fi ) of Ui -homomorphisms of E , into
Ei satisfying (1.5.4.). Let s be a section of E , given by a system ( si ) of sections of Ei
over Ui. Then the system ( fisi ) of sections of E
0 i over^ Ui defines the section^ fs^ (trivial).
The reader may check, as an exercise, how the canonical maps of spaces of sections
considered above in this section, can be made explicit for fibre spaces given by
means of coordinate transformations.
2 Sheaves of sets
Throughout this exposition, we will now use the word “section” for “continuous
section”.
2.1 Sheaves of sets.
Definition 2.1.1. Let X be a space. A sheaf of sets on X (or simply a sheaf) is a fibre
space ( E, X, p ) with base X , satisfying the condition : each point a of E has an open
neighborhood U such that p induces a homeomorphism of U onto an open subset
p ( U ) of X.
This can be expressed by saying that p is an interior map and a local
homeomorphism. It should be kept in mind that, even if X is separated, E is not
supposed separated (and will in most important instances not be separated).
With the notations of definition 2.1.1, let x = p ( a ). If f is a section of E such that fx = a ,
then V = f
− 1 ( U ) ∩ p ( U ) is an open set containing x , and on this neighborhood V of x , f
must coincide with the inverse of the homeomorphism p|U of U onto p ( U ). In particular
Proposition 2.1.1. Two sections of a sheaf E defined in a neighborhood of x and
taking the same value at x coincide in some neighborhood of X.
Corollary : Given two sections of E in an open set V , the set of points where they are
equal is open. (But in general not closed, as would be the case if E were separated !).
2.2 H
0
( A, E ) for arbitrary A ⊂ X.
First let E be an arbitrary fibre space over X. Let A be an arbitrary subset of X ; the
open neighborhoods of A , ordered by ⊃, form an ordered filtering set. To each
element U of this set is associated a set H
0 ( U, E ) : the set of sections of E over U , and
if U ⊃ V ( U and V open neighborhoods of A ), we have a natural map S V U : H
0 ( U, E )
−→ H
0 ( V, E ) (restriction map), with the evident transitivity property S W V S V U = S W U
when U ⊃ V ⊃ W. Therefore we can consider the direct limit of the family of sets
H
0 ( U, E ) for the maps S V U.
Definition 2.2.1. We put H
0 ( A, E ) = lim −→H
0 ( U, E ), ( U ranging over the open neigh
borhoods as explained above). If A = {x} ( x ∈ X ), we simply write H
0 ( x, E ). The
elements of H
0 ( A, E ) are called germs of sections of E in the neighborhood of A.
If A is open, we find of course nothing else but the set of continuous sections of E
over A , already denoted by H
0 ( A, E ). If A ⊃ B , there is a natural map, again noted
S BA of H
0 ( A, E ) into H
0 ( B, E ), (definition left to the reader). When A and B are both
open, this is the usual restriction map (therefore it will in general still be called
restriction map) ; when A is open, then this is the natural homomorphism of H
0 ( A, E )
into the direct limit of all H
0 ( A
0 , E ) corresponding to open neighborhoods A
0 of B. Of
course A ⊃ B ⊃ C implies S CB S BA = S CA.
Let Γ( A, E ) be the set of continuous sections of E over the arbitrary set A ⊃ X , then
the restriction maps H
0 ( U, E ) = Γ( U, E ) −→ Γ( A, E ) ( U , open neighborhood of A )
define a natural map of lim −→H
0 ( U, E ) = H
0 ( A, E ) into Γ( A, E ). In particular, there is a
natural map H
0 ( x, E ) −→ Ex , where Ex is the fibre of x in E (value at x of a germ of
section in a neighborhood of x ). This of course, though frequently an onto-map, will
seldom be one-to-one. However :
Proposition 2.2.1. If E is a sheaf on X , then for x ∈ X , the canonical map H
0 ( x, E ) −→
Ex , is bijective (i.e, one-to-one and onto). If A is any subset of X , then the canonical
map H
0 ( A, E ) −→ Γ( A, E ) is one-to-one ; it is moreover onto if A admits a fundamental
system of paracompact neighborhoods.
The one-to-one parts are contained in Proposition 2.1.1 and its corollary. The first onto
assertion results at once from definition 2.1.1. Now let f be a continuous section of E
over A ; for any x ∈ A , let gx be a continuous section of E on an open neighborhood
Vx of x in X , such that gx ( x ) = f ( x ) (these exist by first part of proposition 2.2.1.).
(2.3.1.) S W V S V U = S W U (if U ⊃ V ⊃ W ) ,
For any x ∈ X , let Ex = lim −→EU , U ranging over the ordered filtering set of open
neighborhoods of x (ordered by ⊃). Let E be the union of the Ex ’s, and p the map of E
into X mapping Ex in x. Define in E a topology as follows : for any f ∈ EU and x ∈ U ,
we consider the canonical image fx ∈ E of f in the direct limit Ex of the sets
E
0 U corresponding^ to^ all^ open^ neighborhoods^ U
0 of x. Let O ( f ) be the set of all
elements fx ∈ E when x ranges over U. When U and f ∈ EU vary, we get a family of
subsets O ( f ) of E , which generate a topology on E. It is easily checked that ( E, X, p )
form a sheaf, that is that p is continuous, interior and a local homeomorphism.
Definition 2.3.1. The sheaf E thus defined is called the sheaf defined by the system of
sets EU and maps S V U.
Consider now an open set U ⊂ X , V -small ; for any f ∈ EU , the map x −→ fx is
clearly a section of the sheaf E , and moreover continuous, which we denote by
e f. We
get thus a natural map f −→
e f of EU into H
0 ( U, E ).
Proposition 2.3.1. In order that f −→
e f be a one-to-one map, it is necessary and suffi
cient that for any open covering ( Ui ) of U , and two elements f, g of EU , S UiU f = S UiU g
for each i implies f = g. In order that f −→
e f be onto, it is necessary and sufficient that
for any open covering ( Ui ) of U , and any system ( fi ) ∈ ∩ EUi satisfying
(2.3.2.) S Ui∩Uj ,Uifi = S Ui∩Uj ,Ujfj when Ui ∩ Uj 6 = ∅
there exists af ∈ EU such that fi = S UiU f for each i.
Corollary. In order that f −→
e f be bijective, it is necessary and sufficient that for any
open covering ( Ui ) of U , the natural map EU −→ ∩ EUi (the components of which are
the maps S UiU ) be a one-to-one map of EU onto the subset of the product of all ( fi )
satisfying condition (2.3.2.).
Proof left to the reader, as well as the proof of the following :
Proposition 2.3.2. Let E be a sheaf on X , consider the system of sets H
0 ( U, E ) and of
restriction maps S V U : H
0 ( U, E ) −→ H
0 ( V, E ) for U ⊃ V ( U, V open sets). Then the
sheaf E
0 defined by these data (definition 2.3.1.) is canonically isomorphic to E , this
isomorphism, transforming for each x ∈ X , E
0 x = lim −→H
0 ( U, E ) = H
0 ( x, E ) into Ex ,
being the isomorphism considered in proposition 2.2.1.
The two preceding propositions show essential equivalence of the notion of sheaf on
the space X , and the notion of a system of sets ( EU ) ( U open ⊂ X ) and of maps S V U
for U ⊃ V , satisfying conditions (2.3.1.) and the condition of corollary of proposition
2.3.1. Both pictures are of importance, the second more intuitive, but the first often
technically more simple.
Exercise. Given a system of sets EU ( U open and V -small) and of homomorphisms
S V U ( U ⊃ V ) satisfying (2.3.1.), prove that if we restrict to those U which are V
0 -small
(where V
0 is an open covering of X finer than V ), the sheaf defined by this new system
is canonically isomorphic to the sheaf defined by the first.
2.4 Permanence properties.
Let E be a sheaf on the space X , and let f be a continuous map of a space X
0 into X ,
then the inverse image of the fibre space E by f (cf 1.2.) is again a sheaf. In particular,
if X
0 ⊂ X , E induces a sheaf on X
0 .
If E is a sheaf on X , F a sheaf on Y , then E × F is a sheaf on X × Y ; therefore, if E
and F are two sheaves on X , then their
fibre-product E × X
this extends to the product of a finite
number of sheaves.
F (cf. 1.3) is again a sheaf ;
Under the conditions of 1.5. suppose that the fibre spaces Ei on the open sets Ui are
sheaves, then the fibre space E obtained by means of coordinate transforms fij is
again a sheaf. This results at once from the more general remark : if E is a fibre space
such that each x ∈ X has a neighborhood U such that E|U be a sheaf, then E is a
sheaf (trivial).
2.5 Subsheaf, quotient sheaf. Homomorphisms of sheaves.
Proposition 2.5.1. Let E be a sheaf on the space X. In order that a subset F of E ,
considered as a fibre space over X , be a sheaf, it is necessary and sufficient that it be
open. In order that the quotient of E by an equivalence relation R compatible with the
fibering, be a sheaf, it is necessary and sufficient that the set of equivalent pairs ( z,
z
0 )
be open in the
fibered product E
× X
E.
subsheaf of E defined by them is nothing else but E
0 .
Now let E, F be two sheaves on X defined by systems ( EU , S V U ) and ( FU , Ψ V U ).
Sup pose given for any U a map fU : EU −→ FU , such that U ⊃ V implies Ψ V U fU =
fU S V U. Then this system of maps defines, for each x ∈ X , a map fx of Ex = lim −→EU
into Fx = lim −→FU
, hence a map f of E into F. It is checked easily (using for instance
pro position 2.5.2.) that f is a homomorphism of E into F. Moreover, f ( E ) is nothing
else but the subsheaf of F defined by the subsets fU ( EU ) of the FU. For any open U ,
the following diagram is commutative.
fu
EU FU
f ∗
H
0 ( U, E ) H
0 ( U, F )
In particular, if the vertical maps are bijective, we see that the maps fU can be
identified with the maps f ∗ : H
0 ( X, E ) −→ H
0 ( X, F ) defined by the homomorphism f.
Conversely, if we start with an arbitrary homomorphism f of E into F , then the
homomorphism defined by the system of maps fU of EU = H
0 ( U, E ) into FU = H
0 ( U, F )
is precisely f.
2.6 Some examples.
a. Constant and locally constant sheaves
Let F be a discrete space, then the trivial fibre space X × F is clearly a sheaf on X ; a
sheaf isomorphic to such a sheaf is called constant. The sections of this sheaf on a
set A ⊂X are the continuous maps of A in the discrete set F , i.e, the maps of A in F
which are locally constant. If for instance A is connected, these reduce to the constant
maps of A into F. Inverse images and products of simple sheaves are simple.
A sheaf E on X is called locally simple, if each x ∈ X has a neighborhood U such that
E|U be simple. Thus a locally simple sheaf on X is nothing else but a covering space
of X in the classical sense (but not restricted of course to be connected). Inverse
images and products of locally simple sheaves in finite number are locally simple.
b. Sheaf of germs of maps. Let X be a space, E a set. Consider for any open U ⊂ X
the set F( U, E ) of all maps of U into E ; if U ⊃ V , we have a natural map of F( U, E )
into F( V, E ), the restriction map. The transitivity condition of section 2.3 is clearly
satisfied, and also the condition of proposition 2.3.1., corollary. Therefore the sets
F( U, E ) can be identified with the sets of sections H
0 ( U, F ) of a well determined sheaf
F, the elements of which are called germs of maps of X into E.
If A ⊂ X , then the elements of H
0 ( A, F) are called germs of maps of a neighborhood
of A into E. If now E is a topological space, we can consider for any U the subset C ( U,
E ) of F( U, E ) of the continuous maps of U into E. As continuity is a condition of local
character, it follows by section 2.5 that the sets C ( U, E ) are the sets of sections of a
well determined subsheaf of F, which is called the sheaf of germs of continuous maps
of X into E. (If we take on E the coarsest topology, we find again the first sheaf.)
Suppose now that E is a fibre space over X , then consider for any U the subset H
0 ( U,
E ) of C ( U, E ) of continuous sections of E. The property of being a section is again of
local character, so we see that the sets H
0 ( U, E ) are sets of sections of a well
determined subsheaf of the sheaf of germs of continuous maps of X into E : the sheaf
of germs of sections of the fibre space E. If this sheaf is denoted by E
e , then H
0 ( A,
E
e ) is nothing else but the set of germs of sections of E in the neighborhood of A , as
defined in definition 2.2.1.
Of course, specializing the spaces X and E , we can define a great number of other
sub sheaves of the sheaf of germs of maps of X into E (germs of differentiable maps,
germs of analytic maps, germs of maps which are L
P etc.).
c. Sheaf of germs of homomorphisms of a fibre space into another.
Let E and F be two fibre spaces over X , and for any open U ⊂ X let HU be the set of
homomorphisms of E|U into F|U. If V is an open set contained in U , there is an
evident natural map of restriction HU −→ HV. The condition of transitivity as well as
the condition of proposition 2.3.1. corollary, are satisfied, so that the sets HU appear
as the sets H
0 ( U, H ) of sections of a well determined sheaf on X , the elements of
which are called germs of homomorphisms of E into F. A section of this sheaf over X
is a homomorphism of E into F.
d. Sheaf of germs of subsets.
Let X be a space, for any open set U ⊂ X let P ( U ) be the set of subsets of U. If U ⊃
V , consider the map A −→ A∩V of P ( U ) into P ( V ). Clearly the conditions of
transitivity, and of proposition 2.3.1. corollary, are satisfied, so that the sets P ( U )
appear as the sets H
0 ( U, P ( X )) of sections of a well determined sheaf on X , the
elements of which are called germs of sets in X. Any condition of a local character on
subsets of X defines a subsheaf of P ( X ), for instance the sheaf of germs of closed
sets (corresponding to the relatively closed sets in U ), or if X is an analytic manifold,
the sheaf of germs of analytic sets, etc.
