## Chapter 4

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 $ℱ$ on $X$ is the assignment of an abelian group $ℱ\left(U\right)$ to each open set $U$ in $X,$ together with homomorphisms (often called restriction homomorphisms) $ℱ\left(U\right)\to ℱ\left(V\right)$ defined whenever $U\supseteq V,$ such that $ℱ\left(U\right)\to ℱ\left(U\right)$ is the identity map, and that the composition $ℱ\left(U\right)\to ℱ\left(V\right)\to ℱ\left(W\right)$ (where $U\supseteq V\supseteq W\text{)}$ is the same as the homomorphism $ℱ\left(U\right)\to ℱ\left(W\right)\text{.}$ (Think of the elements of $ℱ\left(U\right)$ as functions on $U\text{.)}$

Another way of saying the same thing is as follows. Let $ℂ\left(X\right)$ be the category whose objects are the open sets in $X$ and whose only morphisms are inclusions of open sets. Then a presheaf $ℱ$ is just a contravariant functor from the category $ℂ\left(X\right)$ into the category $\text{Ab}$ 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 $ℱ$ is a sheaf if it satisfies the following condition:

For each open set $U$ in $X$ and each open covering $\left({U}_{\alpha }\right)$ of a $U,$ and each family $\left({s}_{\alpha }\right)$ such that ${s}_{\alpha }\in ℱ\left({U}_{\alpha }\right)$ and ${s}_{\alpha },{s}_{\beta }$ have the same restriction to $ℱ\left({U}_{\alpha }\cap {U}_{\beta }\right)$ for all $\alpha ,\beta ,$ there is a unique $s\in ℱ\left(U\right)$ whose restriction to ${U}_{\alpha }$ is ${s}_{\alpha },$ for all $\alpha \text{.}$

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

$A⟶B ⇉v2v1C$

is said to be exact if $u$ maps $A$ one-one onto the set of all $x\in B$ such that ${v}_{1}\left(x\right)={v}_{2}\left(x\right)\text{.}$ Then $ℱ$ is a sheaf if and only if, for each open set $U$ in $X$ and each open covering $\left({U}_{\alpha }\right)$ of $U,$ the diagram

$ℱ(U)⟶ ∏αℱ(Uα) ⇉∏α,β ℱ(Uα∩Uβ)$

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

### Stalks

Let $ℱ$ be a presheaf (say of abelian groups) on $X$ and let $x$ be a point of $X\text{.}$ Then the direct limit $\underset{\to }{\text{lim}}ℱ\left(U\right),$ where $U$ runs through all open neighbourhoods of $x$ in $X,$ is called the stalk of $ℱ$ at $x$ and is denoted by ${ℱ}_{x}\text{.}$ Thus an element ${s}_{x}\in {ℱ}_{x}$ is represented by some $s\in ℱ\left(U\right),$ where $U$ is some open neighbourhood of $x$ in $X,$ and two elements $s\in ℱ\left(U\right)$ and $s\prime \in ℱ\left(U\prime \right)$ represent the same element of ${ℱ}_{x}$ if and only if there is an open neighbourhood ${U}^{\prime \prime }$ of $x$ contained in $U\cap U\prime$ such that the restrictions of $s$ and $s\prime$ to ${U}^{\prime \prime }$ are the same.

If $U$ is any open set in $X$ and if $x$ is any point of $U,$ we have a homomorphism $ℱ\left(U\right)\to {ℱ}_{x}\text{.}$ If $s\in ℱ\left(U\right)$ we denote the image of $s$ under this homomorphism by ${s}_{x}\text{.}$

### The sheaf associated with a presheaf

Let $ℱ$ be a presheaf on $X$ and let $E$ denote the disjoint union, or sum, of the stalks ${ℱ}_{x}\text{;}$ then $E$ has a natural projection $p$ onto $X,$ namely the fibre ${p}^{-1}\left(x\right)$ is the stalk ${ℱ}_{x}$ of $ℱ$ at $x\text{.}$ For each open set $U$ in $X$ and each $s\in ℱ\left(U\right),$ let $\stackrel{\sim }{s}\left(x\right)$ denote ${s}_{x}\text{;}$ then $\stackrel{\sim }{s}:U\to E$ is a section of $E$ over $U,$ i.e., $p\circ \stackrel{\sim }{s}$ is the identity map of $U\text{.}$ We can make $E$ into a topological space by giving $E$ the coarsest topology for which all the mappings $\stackrel{\sim }{s}$ are continuous: this means that a set $W$ is open in $E$ if and only if, for each open $U\subseteq X$ and each $s\in ℱ\left(U\right),$ the set of points $x\in U$ such that $\stackrel{\sim }{s}\left(x\right)\in W$ form an open set in $X\text{.}$

Let $\stackrel{\sim }{ℱ}\left(U\right)$ denote the set of continuous, sections of $E$ over $U\text{.}$ Then an element of $\stackrel{\sim }{ℱ}\left(U\right)$ is a family ${\left({s}_{x}^{\prime }\right)}_{x\in U\prime }$ where ${s}_{x}^{\prime }\in {ℱ}_{x}$ for all $x\in U,$ such that for each $x\in U$ there is an open neighbourhood $V$ of $x,$ contained in $U,$ and an element $s\in ℱ\left(V\right)$ such that ${s}_{y}^{\prime }={s}_{y}$ for all $y\in V\text{.}$ It is easily checked that

Lemma (4.1). $\stackrel{\sim }{ℱ}$ is a sheaf.

If $ℱ,𝒢$ are presheaves on $X,$ a homomorphism $\phi :ℱ\to 𝒢$ is a family of homomorphisms $\phi \left(U\right):ℱ\left(U\right)\to 𝒢\left(U\right)$ for each open set $U$ in $X,$ which are compatible with the restriction homomorphisms in $ℱ$ and $𝒢\text{:}$ that is, whenever $U,V$ are open in $X$ and $U\supseteq V,$ the diagram

$ℱ(U) ⟶φ(U) 𝒢(U) ↓ ↓ ℱ(V) ⟶φ(V) 𝒢(V)$

(in which the vertical arrows are restrictions) is commutative. If we regard $ℱ,𝒢$ as contravariant functors on the category $ℂ\left(X\right),$ then $\phi$ is just a morphism (or natural transformation) of functors.

In particular, let $ℱ$ be a presheaf on $X,$ $ℱ$ the associated sheaf (4.1). For each open set $U$ in $X$ and each $s\in ℱ\left(U\right),$ the family ${\left({s}_{x}\right)}_{x\in U}$ is an element of $ℱ\left(U\right),$ so that we have a homomorphism $ℱ\to \stackrel{\sim }{ℱ}\text{.}$

Lemma (4.2). $ℱ\to \stackrel{\sim }{ℱ}$ is an isomorphism if and only if $ℱ$ is a sheaf.

If $ℱ$ is a sheaf, we shall often use the notation $\Gamma \left(U,ℱ\right)$ instead of $ℱ\left(U\right)\text{.}$

### Restriction of a presheaf to an open set

Let $ℱ$ be a presheaf on $X,$ and let $U$ be an open set in $X\text{.}$ Then the $ℱ\left(V\right)$ for which $V\subseteq U$ form a presheaf on $U,$ called the restriction of $ℱ$ to $U$ and denoted by $ℱ|U\text{.}$ If $ℱ$ is a sheaf, so is $ℱ|U$ (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 $ℬ$ be a basis of open sets in $X\text{.}$ A presheaf on $ℬ$ (say a presheaf of abelian groups) is the assignment of an abelian group $ℱ\left(U\right)$ to each $U\in ℬ,$ together with restriction homomorphisms $ℱ\left(U\right)\to ℱ\left(V\right)$ whenever $U,V\in ℬ$ and $U\supseteq V,$ satisfying the same conditions as before.

From a presheaf $ℱ$ on $ℬ$ we can construct a presheaf $ℱ\prime$ on $X$ in the previous sense: if $U$ is any open set in $X,$ then $ℱ\prime \left(U\right)$ is defined to be the inverse limit $\underset{←}{\text{lim}} ℱ\left(V\right),$ taken over all $V\in ℬ$ such that $V\subseteq U\text{.}$ Explicitly, an etement $s\prime \in ℱ\left(U\right)$ is a family ${\left({s}_{V}\right)}_{V\in ℬ,V\subseteq U},$ such that if $V,W\in ℬ$ and $U\supseteq V\supseteq W,$ then the restriction of ${s}_{V}$ to $W$ is ${s}_{W}\text{.}$ If $U\in ℬ,$ then $ℱ\prime \left(U\right)$ is canonically isomorphic to $ℱ\left(U\right)\text{.}$

Lemma (4.3). With the above notation, $ℱ\prime$ is a sheaf on $X$ if and only if $ℱ$ satisfies the followinq condition: for each $U\in ℬ$ and each covering $\left({U}_{\alpha }\right)$ of $U$ by sets belonging to $ℬ,$ the diagram

$ℱ(U)⟶ ∏αℱ(Uα) ⇉∏α,β ∏V∈ℬV⊆Uα∩Ubeta ℱ(V)$

is exact: that is, if ${s}_{\alpha }\in ℱ\left({U}_{\alpha }\right)$ are such that the restrictions of ${s}_{\alpha }$ and ${s}_{\beta }$ to $V$ are the same, for all pairs $\alpha ,\beta$ and all $V\subseteq {U}_{\alpha }\cap {U}_{\beta }$ $\left(V\in ℬ\right),$ then there is a unique $s\in ℱ\left(U\right)$ whose restriction to ${U}_{\alpha }$ is ${s}_{\alpha }$ for all $\alpha \text{.}$

The stalk ${ℱ}_{x}^{\prime }$ of $ℱ\prime$ at $x$ is equal to $\underset{←}{\text{lim}} ℱ\left(U\right),$ where $U$ runs through all sets of $ℬ$ which contain $x,$ because these sets are cofinal in the set of all open neighbourhoods of $x\text{.}$

### Ringed spaces

A ringed space (espace annalé) is a pair $\left(X,{𝒪}_{X}\right)$ where $X$ is a topological space and ${𝒪}_{X}$ 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 $𝒪\left(U\right)$ denote the ring of all holomorphic functions defined on $U\text{.}$ Then $𝒪$ is a sheaf of rings on $X,$ so that a complex manifold may be regarded as a ringed space $\left(X,𝒪\right)\text{.}$ Similarly for differentiable manifolds, algebraic varieties over a field, etc.

A morphism of ringed spaces $\left(X,{𝒪}_{X}\right)\to \left(Y,{𝒪}_{Y}\right)$ is a pair $\left(\psi ,\theta \right),$ where $\psi$ is a continuous map from $X$ to $Y,$ and $\theta$ maps ${𝒪}_{Y}$ to ${𝒪}_{X}\text{;}$ precisely, $\theta$ assigns to each open set $V$ in $Y$ a ring homomorphism $\theta \left(V\right):\Gamma \left(V,{𝒪}_{Y}\right)\to \Gamma \left({\psi }^{-1}\left(V\right),{𝒪}_{X}\right),$ compatible with the restriction homomorphisms: that is to say, whenever $V\supseteq V\prime$ 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 $x\in X,$ $\theta$ 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".