Last update: 3 June 2013
At this stage we need little more than the definitions. Let be a topological space. A presheaf of abelian groups on is the assignment of an abelian group to each open set in together with homomorphisms (often called restriction homomorphisms) defined whenever such that is the identity map, and that the composition (where is the same as the homomorphism (Think of the elements of as functions on
Another way of saying the same thing is as follows. Let be the category whose objects are the open sets in and whose only morphisms are inclusions of open sets. Then a presheaf is just a contravariant functor from the category into the category of abelian groups. Put this way, it is clear how to define a presheaf on with values in any given category: for example, presheaves of rings, modules etc.
A presheaf is a sheaf if it satisfies the following condition:
For each open set in and each open covering of a and each family such that and have the same restriction to for all there is a unique whose restriction to is for all
Another way of putting this is as follows. A diagram of sets and mappings
is said to be exact if maps one-one onto the set of all such that Then is a sheaf if and only if, for each open set in and each open covering of the diagram
(in which the maps are products of restriction homomorphisms) is exact.
Let be a presheaf (say of abelian groups) on and let be a point of Then the direct limit where runs through all open neighbourhoods of in is called the stalk of at and is denoted by Thus an element is represented by some where is some open neighbourhood of in and two elements and represent the same element of if and only if there is an open neighbourhood of contained in such that the restrictions of and to are the same.
If is any open set in and if is any point of we have a homomorphism If we denote the image of under this homomorphism by
Let be a presheaf on and let denote the disjoint union, or sum, of the stalks then has a natural projection onto namely the fibre is the stalk of at For each open set in and each let denote then is a section of over i.e., is the identity map of We can make into a topological space by giving the coarsest topology for which all the mappings are continuous: this means that a set is open in if and only if, for each open and each the set of points such that form an open set in
Let denote the set of continuous, sections of over Then an element of is a family where for all such that for each there is an open neighbourhood of contained in and an element such that for all It is easily checked that
Lemma (4.1). is a sheaf.
If are presheaves on a homomorphism is a family of homomorphisms for each open set in which are compatible with the restriction homomorphisms in and that is, whenever are open in and the diagram
(in which the vertical arrows are restrictions) is commutative. If we regard as contravariant functors on the category then is just a morphism (or natural transformation) of functors.
In particular, let be a presheaf on the associated sheaf (4.1). For each open set in and each the family is an element of so that we have a homomorphism
Lemma (4.2). is an isomorphism if and only if is a sheaf.
If is a sheaf, we shall often use the notation instead of
Let be a presheaf on and let be an open set in Then the for which form a presheaf on called the restriction of to and denoted by If is a sheaf, so is (obvious from the definitions).
We shall need a slight variant of the above notion of a presheaf. Let be a topological space and let be a basis of open sets in A presheaf on (say a presheaf of abelian groups) is the assignment of an abelian group to each together with restriction homomorphisms whenever and satisfying the same conditions as before.
From a presheaf on we can construct a presheaf on in the previous sense: if is any open set in then is defined to be the inverse limit taken over all such that Explicitly, an etement is a family such that if and then the restriction of to is If then is canonically isomorphic to
Lemma (4.3). With the above notation, is a sheaf on if and only if satisfies the followinq condition: for each and each covering of by sets belonging to the diagram
is exact: that is, if are such that the restrictions of and to are the same, for all pairs and all then there is a unique whose restriction to is for all
The stalk of at is equal to where runs through all sets of which contain because these sets are cofinal in the set of all open neighbourhoods of
A ringed space (espace annalé) is a pair where is a topological space and is a sheaf of rings on called the structure sheaf of the ringed space.
Example. Let be a complex manifold, and for each open set in let denote the ring of all holomorphic functions defined on Then is a sheaf of rings on so that a complex manifold may be regarded as a ringed space Similarly for differentiable manifolds, algebraic varieties over a field, etc.
A morphism of ringed spaces is a pair where is a continuous map from to and maps to precisely, assigns to each open set in a ring homomorphism compatible with the restriction homomorphisms: that is to say, whenever are open sets in the diagram
is commutative. For each then induce's a homomorphism of the stalks
by taking direct limits.
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".