Last update: 3 June 2013

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

Another way of saying the same thing is as follows. Let $\u2102\left(X\right)$ be the category whose objects are the open sets in $X$ and whose only morphisms are inclusions of open sets. Then a presheaf $\mathcal{F}$ is just a contravariant functor from the category $\u2102\left(X\right)$ 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 $\mathcal{F}$ 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
${s}_{\alpha}\in \mathcal{F}\left({U}_{\alpha}\right)$
and ${s}_{\alpha},{s}_{\beta}$ have the same restriction to
$\mathcal{F}({U}_{\alpha}\cap {U}_{\beta})$
for all $\alpha ,\beta ,$ there is a *unique*
$s\in \mathcal{F}\left(U\right)$ 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

$$A\u27f6B\underset{{v}_{2}}{\overset{{v}_{1}}{\rightrightarrows}}C$$
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 $\mathcal{F}$ 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

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

Let $\mathcal{F}$ be a presheaf (say of abelian groups) on $X$ and let $x$ be a point of $X\text{.}$
Then the direct limit $\underrightarrow{\text{lim}}\mathcal{F}\left(U\right),$
where $U$ runs through all open neighbourhoods of $x$ in $X,$ is called the
*stalk* of $\mathcal{F}$ at $x$ and is denoted by ${\mathcal{F}}_{x}\text{.}$
Thus an element ${s}_{x}\in {\mathcal{F}}_{x}$ is represented by some
$s\in \mathcal{F}\left(U\right),$ where $U$ is some open
neighbourhood of $x$ in $X,$ and two elements
$s\in \mathcal{F}\left(U\right)$ and
$s\prime \in \mathcal{F}\left(U\prime \right)$
represent the same element of ${\mathcal{F}}_{x}$ 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 $\mathcal{F}\left(U\right)\to {\mathcal{F}}_{x}\text{.}$ If $s\in \mathcal{F}\left(U\right)$ we denote the image of $s$ under this homomorphism by ${s}_{x}\text{.}$

Let $\mathcal{F}$ be a presheaf on $X$ and let $E$ denote the disjoint union, or sum, of the stalks
${\mathcal{F}}_{x}\text{;}$ then $E$ has a natural projection $p$ onto
$X,$ namely the fibre ${p}^{-1}\left(x\right)$
is the stalk ${\mathcal{F}}_{x}$ of $\mathcal{F}$ at $x\text{.}$ For each
open set $U$ in $X$ and each $s\in \mathcal{F}\left(U\right),$
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
$s\in \mathcal{F}\left(U\right),$ 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 $\stackrel{\sim}{\mathcal{F}}\left(U\right)$ denote the set of continuous, sections of $E$ over $U\text{.}$ Then an element of $\stackrel{\sim}{\mathcal{F}}\left(U\right)$ is a family ${\left({s}_{x}^{\prime}\right)}_{x\in U\prime}$ where ${s}_{x}^{\prime}\in {\mathcal{F}}_{x}$ 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 $s\in \mathcal{F}\left(V\right)$ such that ${s}_{y}^{\prime}={s}_{y}$ for all $y\in V\text{.}$ It is easily checked that

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

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

(in which the vertical arrows are restrictions) is commutative. If we regard $\mathcal{F},\mathcal{G}$ as contravariant functors on the category $\u2102\left(X\right),$ then $\phi $ is just a morphism (or natural transformation) of functors.

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

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

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

Let $\mathcal{F}$ be a presheaf on $X,$ and let $U$ be an open set in $X\text{.}$
Then the $\mathcal{F}\left(V\right)$ for which
$V\subseteq U$ form a presheaf on $U,$ called the *restriction* of
$\mathcal{F}$ to $U$ and denoted by $\mathcal{F}|U\text{.}$ If
$\mathcal{F}$ is a sheaf, so is $\mathcal{F}|U$ (obvious from the definitions).

We shall need a slight variant of the above notion of a presheaf. Let $X$ be a topological space and let $\mathcal{B}$ be a basis
of open sets in $X\text{.}$ A *presheaf on* $\mathcal{B}$ (say a presheaf of abelian groups) is the
assignment of an abelian group $\mathcal{F}\left(U\right)$ to each
$U\in \mathcal{B},$ together with restriction homomorphisms
$\mathcal{F}\left(U\right)\to \mathcal{F}\left(V\right)$
whenever $U,V\in \mathcal{B}$ and
$U\supseteq V,$ satisfying the same conditions as before.

From a presheaf $\mathcal{F}$ on $\mathcal{B}$ we can construct a presheaf $\mathcal{F}\prime $ on $X$ in the previous sense: if $U$ is any open set in $X,$ then $\mathcal{F}\prime \left(U\right)$ is defined to be the inverse limit $\underleftarrow{\text{lim}}\hspace{0.17em}\mathcal{F}\left(V\right),$ taken over all $V\in \mathcal{B}$ such that $V\subseteq U\text{.}$ Explicitly, an etement $s\prime \in \mathcal{F}\left(U\right)$ is a family ${\left({s}_{V}\right)}_{V\in \mathcal{B},V\subseteq U},$ such that if $V,W\in \mathcal{B}$ and $U\supseteq V\supseteq W,$ then the restriction of ${s}_{V}$ to $W$ is ${s}_{W}\text{.}$ If $U\in \mathcal{B},$ then $\mathcal{F}\prime \left(U\right)$ is canonically isomorphic to $\mathcal{F}\left(U\right)\text{.}$

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

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

The *stalk* ${\mathcal{F}}_{x}^{\prime}$ of
$\mathcal{F}\prime $ at $x$ is equal to
$\underleftarrow{\text{lim}}\hspace{0.17em}\mathcal{F}\left(U\right),$
where $U$ runs through all sets of $\mathcal{B}$ which contain $x,$ because these
sets are cofinal in the set of all open neighbourhoods of $x\text{.}$

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

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

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

$${\theta}_{x}^{\#}:{\mathcal{O}}_{Y,\psi \left(x\right)}\u27f6{\mathcal{O}}_{X,x}$$by taking direct limits.

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