Chapter 4

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $ℱ(U_{\alpha }\cap U_{\beta })$ 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

$$A⟶B\underset{v_2}{\overset{v_1}{\rightrightarrows }}C$$

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

$$ℱ\left(U\right)⟶\prod _{\alpha }ℱ\left(U_{\alpha }\right)\rightrightarrows \prod _{\alpha ,\beta }ℱ(U_{\alpha }\cap U_{\beta })$$

(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 $\underrightarrow{\text{lim}}ℱ\left(U\right),$ 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

$$\begin{array}{ccc}ℱ\left(U\right)& \stackrel{\phi \left(U\right)}{⟶}& 𝒢\left(U\right)\\ \downarrow & & \downarrow \\ ℱ\left(V\right)& \underset{\phi \left(V\right)}{⟶}& 𝒢\left(V\right)\end{array}$$

(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 $\Gamma (U,ℱ)$ 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 $\underleftarrow{\text{lim}}\hspace{0.17em}ℱ\left(V\right),$ 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

$$ℱ\left(U\right)⟶\prod _{\alpha }ℱ\left(U_{\alpha }\right)\rightrightarrows \prod _{\alpha ,\beta }\prod _{\underset{V\subseteq U_{\alpha }\cap U_{}}{V\in ℬ}}ℱ\left(V\right)$$

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 $V\subseteq U_{\alpha }\cap U_{\beta }$ $(V\in ℬ),$ 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 $\underleftarrow{\text{lim}}\hspace{0.17em}ℱ\left(U\right),$ 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 $(X,{𝒪}_X)$ 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 $(X,𝒪)\text{.}$ Similarly for differentiable manifolds, algebraic varieties over a field, etc.

A morphism of ringed spaces $(X,{𝒪}_X)\to (Y,{𝒪}_Y)$ is a pair $(\psi ,\theta ),$ 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 $\theta \left(V\right):\Gamma (V,{𝒪}_Y)\to \Gamma ({\psi }^{-1}\left(V\right),{𝒪}_X),$ compatible with the restriction homomorphisms: that is to say, whenever $V\supseteq V\prime$ are open sets in $Y,$ the diagram

$$\begin{array}{ccc}\Gamma (V,{𝒪}_Y)& \stackrel{\theta \left(V\right)}{⟶}& \Gamma ({\psi }^{-1}\left(V\right),{𝒪}_X)\\ \downarrow & & \downarrow \\ \Gamma (V\prime ,{𝒪}_Y)& \underset{\theta \left(V\prime \right)}{⟶}& \Gamma ({\psi }^{-1}\left(V\prime \right),{𝒪}_X)\end{array}$$

is commutative. For each $x\in X,$ $\theta$ then induce's a homomorphism of the stalks

$${\theta }_x^{\#}:{𝒪}_{Y,\psi \left(x\right)}⟶{𝒪}_{X,x}$$

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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