Chapter 4

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 3 June 2013

Presheaves and Sheaves

At this stage we need little more than the definitions. Let X be a topological space. A presheaf of abelian groups on X is the assignment of an abelian group (U) to each open set U in X, together with homomorphisms (often called restriction homomorphisms) (U)(V) defined whenever UV, such that (U)(U) is the identity map, and that the composition (U)(V)(W) (where UVW) is the same as the homomorphism (U)(W). (Think of the elements of (U) as functions on U.)

Another way of saying the same thing is as follows. Let (X) be the category whose objects are the open sets in X and whose only morphisms are inclusions of open sets. Then a presheaf is just a contravariant functor from the category (X) into the category Ab of abelian groups. Put this way, it is clear how to define a presheaf on X 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 U in X and each open covering (Uα) of a U, and each family (sα) such that sα(Uα) and sα,sβ have the same restriction to (UαUβ) for all α,β, there is a unique s(U) whose restriction to Uα is sα, for all α.

Another way of putting this is as follows. A diagram of sets and mappings

AB v2v1C

is said to be exact if u maps A one-one onto the set of all xB such that v1(x)=v2(x). Then is a sheaf if and only if, for each open set U in X and each open covering (Uα) of U, the diagram

(U) α(Uα) α,β (UαUβ)

(in which the maps are products of restriction homomorphisms) is exact.


Let be a presheaf (say of abelian groups) on X and let x be a point of X. Then the direct limit lim(U), where U runs through all open neighbourhoods of x in X, is called the stalk of at x and is denoted by x. Thus an element sxx is represented by some s(U), where U is some open neighbourhood of x in X, and two elements s(U) and s(U) represent the same element of x if and only if there is an open neighbourhood U of x contained in UU such that the restrictions of s and s to U are the same.

If U is any open set in X and if x is any point of U, we have a homomorphism (U)x. If s(U) we denote the image of s under this homomorphism by sx.

The sheaf associated with a presheaf

Let be a presheaf on X and let E denote the disjoint union, or sum, of the stalks x; then E has a natural projection p onto X, namely the fibre p-1(x) is the stalk x of at x. For each open set U in X and each s(U), let s(x) denote sx; then s:UE is a section of E over U, i.e., ps is the identity map of U. We can make E into a topological space by giving E the coarsest topology for which all the mappings s are continuous: this means that a set W is open in E if and only if, for each open UX and each s(U), the set of points xU such that s(x)W form an open set in X.

Let (U) denote the set of continuous, sections of E over U. Then an element of (U) is a family (sx)xU where sxx for all xU, such that for each xU there is an open neighbourhood V of x, contained in U, and an element s(V) such that sy=sy for all yV. It is easily checked that

Lemma (4.1). is a sheaf.

If ,𝒢 are presheaves on X, a homomorphism φ:𝒢 is a family of homomorphisms φ(U):(U)𝒢(U) for each open set U in X, which are compatible with the restriction homomorphisms in and 𝒢: that is, whenever U,V are open in X and UV, the diagram

(U) φ(U) 𝒢(U) (V) φ(V) 𝒢(V)

(in which the vertical arrows are restrictions) is commutative. If we regard ,𝒢 as contravariant functors on the category (X), then φ is just a morphism (or natural transformation) of functors.

In particular, let be a presheaf on X, the associated sheaf (4.1). For each open set U in X and each s(U), the family (sx)xU is an element of (U), 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 Γ(U,) instead of (U).

Restriction of a presheaf to an open set

Let be a presheaf on X, and let U be an open set in X. Then the (V) for which VU form a presheaf on U, called the restriction of to U and denoted by |U. If is a sheaf, so is |U (obvious from the definitions).

Presheaf on a base of open sets

We shall need a slight variant of the above notion of a presheaf. Let X be a topological space and let be a basis of open sets in X. A presheaf on (say a presheaf of abelian groups) is the assignment of an abelian group (U) to each U, together with restriction homomorphisms (U)(V) whenever U,V and UV, satisfying the same conditions as before.

From a presheaf on we can construct a presheaf on X in the previous sense: if U is any open set in X, then (U) is defined to be the inverse limit lim(V), taken over all V such that VU. Explicitly, an etement s(U) is a family (sV)V,VU, such that if V,W and UVW, then the restriction of sV to W is sW. If U, then (U) is canonically isomorphic to (U).

Lemma (4.3). With the above notation, is a sheaf on X if and only if satisfies the followinq condition: for each U and each covering (Uα) of U by sets belonging to , the diagram

(U) α(Uα) α,β VVUαUbeta (V)

is exact: that is, if sα(Uα) are such that the restrictions of sα and sβ to V are the same, for all pairs α,β and all VUαUβ (V), then there is a unique s(U) whose restriction to Uα is sα for all α.

The stalk x of at x is equal to lim(U), where U runs through all sets of which contain x, because these sets are cofinal in the set of all open neighbourhoods of x.

Ringed spaces

A ringed space (espace annalé) is a pair (X,𝒪X) where X is a topological space and 𝒪X is a sheaf of rings on X, called the structure sheaf of the ringed space.

Example. Let X be a complex manifold, and for each open set U in X let 𝒪(U) denote the ring of all holomorphic functions defined on U. Then 𝒪 is a sheaf of rings on X, so that a complex manifold may be regarded as a ringed space (X,𝒪). Similarly for differentiable manifolds, algebraic varieties over a field, etc.

A morphism of ringed spaces (X,𝒪X)(Y,𝒪Y) is a pair (ψ,θ), where ψ is a continuous map from X to Y, and θ maps 𝒪Y to 𝒪X; precisely, θ assigns to each open set V in Y a ring homomorphism θ(V):Γ(V,𝒪Y)Γ(ψ-1(V),𝒪X), compatible with the restriction homomorphisms: that is to say, whenever VV are open sets in Y, the diagram

Γ(V,𝒪Y) θ(V) Γ(ψ-1(V),𝒪X) Γ(V,𝒪Y) θ(V) Γ(ψ-1(V),𝒪X)

is commutative. For each xX, θ then induce's a homomorphism of the stalks

θx#: 𝒪Y,ψ(x) 𝒪X,x

by taking direct limits.

Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".

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