Sheaves and Ringed Spaces

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

Last updates: 9 September 2012

Sheaves and ringed spaces

Let $X$ be a topological space with topology $𝒯$. View $𝒯$ as a category with morphisms inclusions of open sets $U\subseteq V$.

Let $𝒜$ be the category of commutative rings with identity.

• A sheaf on $X$ is a contravariant functor $ℱ:𝒯\to 𝒜$ such that if $U\in 𝒯$ and $𝒮\subseteq 𝒯$ is an open cover of $U$ then $$ℱ\left(U\right)\cong \{\left(s_{\alpha }\right)\in \prod _{U_{\alpha }\in 𝒮}ℱ\left(U_{\alpha }\right)|{\varphi }_{\alpha }^{\alpha \beta }\left(s_{\alpha }\right)={\varphi }_{\beta }^{\alpha \beta }\left(s_{\beta }\right)\},$$ where $${\varphi }_{\alpha }^{\alpha \beta }:ℱ\left(U_{\alpha }\right)\to ℱ(U_{\alpha }\cap U_{\beta }),\text{for}{U}_{\alpha },U_{\beta }\in 𝒮,$$ is $ℱ\left(\right(U_{\alpha }\cap U_{\beta })\subseteq U_{\alpha })$, and the isomorphism between the left hand side and the right and side is the function ${\prod }_{U_{\alpha }\in 𝒮}{\varphi }_{\alpha }$, where ${\varphi }_{\alpha }$ is the restriction map $ℱ\left(\right(U_{\alpha }\cap U)\subseteq U)$.
• A morphism of sheaves is a morphism of functors.
• A ringed space is a pair $(X,{𝒪}_X)$ where $X$ is a topological space and ${𝒪}_X$ is a sheaf of rings on $X$.
• Let $(X,{𝒪}_X)$ be a ringed space and let $x\in X$. The stalk of ${𝒪}_X$ at $x\in X$ is $${𝒪}_{X,x}=\mathrm{ind}\underset{U}{\lim }{𝒪}_X\left(U\right),$$ where the limit is over all neighbourhoods $U$ of $x$.

Another way to state the overlap condition in the definition of a sheaf is that if $\left\{U_{\alpha }\right\}$ is an open cover of $U$ and $f_{\alpha }\in {𝒪}_X\left(U_{\alpha }\right)$ are such that $$f_{\alpha }{|}_{U_{\alpha }\cap U_{\beta }}=f_{\beta }{|}_{U_{\alpha }\cap U_{\beta }},\text{for all}\alpha ,\beta ,$$ then there is a unique $f\in {𝒪}_X\left(U\right)$ such that $f_{\alpha }=f{|}_{U_{\alpha }}$ for all $\alpha$.

Yet another way of stating the overlap condition in the definition of a sheaf is to say that the sequence $$0\to {𝒪}_X\left(U\right)\stackrel{i}{\to }\prod _{\alpha }{𝒪}_X\left(U_{\alpha }\right)\underset{\underset{k}{\to }}{\overset{\stackrel{j}{\to }}{\phantom{-}}}\prod _{\alpha ,\beta }{𝒪}_X(U_{\alpha }\cap U_{\beta })$$ is exact, where

1. $i$ is the map induced by the inclusions $U_{\alpha }\to U$,
2. $j$ is the map induced by the inclusions $U_{\alpha }\cap U_{\beta }\to U_{\alpha }$,
3. $k$ is the map induced by the inclusions $U_{\alpha }\cap U_{\beta }\to U_{\beta }$,

and exactness of the sequence means $\mathrm{im}i=\ker (j-k)$.

Direct and inverse image

Let $f:X\to Y$ be continuous.

The direct image

$f_{✶}:\text{Sheaves}\left(X\right)\to \text{Sheaves}\left(Y\right)$ is given by $\left(f_{✶}ℱ\right)\left(V\right)=ℱ\left(f^{-1}\left(V\right)\right)$,

for $V$ an open set of $Y$.

The inverse image $f^{-1}:\text{Sheaves}\left(Y\right)\to \text{Sheaves}\left(X\right)$ is the adjoint of $f_{✶}$,

${\text{Hom}}_{\text{Sh}\left(X\right)}(f^{-1}𝒢,ℱ)\simeq {\text{Hom}}_{\text{Sh}\left(Y\right)}(𝒢,f_{✶}ℱ)$,

constructed by

$f^{-1}𝒢=\text{sheafification}\left(ℱ\right)$, where $ℱ\left(U\right)={\displaystyle \underset{V\supseteq ℱ\left(U\right)}{\text{lim}}}𝒢\left(V\right)$.

The global sections functor $\Gamma :\text{Sheaves}\left(X\right)\to \text{AbelGps}$ is the direct image of $f:X\to \text{pt}$,

$\Gamma =f_{✶}$, where $f:X\to \text{pt}$.

The constant sheaf functor $A:\text{AbelGps}\to \text{Sheaves}\left(X\right)$ is the inverse image of $f:X\to \text{pt}$,

$A=f^{-1}$, where $f:X\to \text{pt}$.

The skyscraper sheaf functor ${\left({\iota }_X\right)}_{✶}:\text{AbelGps}\to \text{Sheaves}\left(X\right)$ is the direct image of ${\iota }_X:\text{pt}\to X$,

${\left({\iota }_X\right)}_{✶}$, where $\begin{array}{rcl}{\iota }_X:\text{pt}& \to & X\\ ✶& \mapsto & X\end{array}$

The stalk functor ${\iota }_X^{-1}:\text{Sheaves}\to \text{AbelGps}$ is the inverse image of ${\iota }_X:\text{pt}\to X$

${\iota }_X^{-1}\left(ℱ\right)={ℱ}_X$ where $\begin{array}{rcl}{\iota }_X:\text{pt}& \to & X\\ ✶& \mapsto & X\end{array}$

HW 1: Show that the global sections of a sheaf $ℱ$ on $X$ is the ring $\Gamma \left(ℱ\right)=ℱ\left(X\right)$.

HW 2: Show that the constant sheaf on $X$ with values in $A$ is given by

$A\left(U\right)=\{f:U\to A\mid \text{If}a\in A\text{then}{f}^{-1}\left(a\right)\text{is open in}U\}$

HW 3: Show that the skyscraper sheaf at $x$ is given by

$A^x\left(V\right)=\{\begin{array}{ccc}A,& & \text{if}x\in U\\ 0,& & \text{otherwise},\end{array}$

HW 4: Show that the stalk of $ℱ$ at $x$ is given by

${\displaystyle {ℱ}_x=\underset{V\supseteq \left\{x\right\}}{\text{lim}}ℱ\left(V\right)}$

Sheaves, presheaves and sheafification

The inclusion, (or restriction, or forgetful) functor is

$\begin{array}{ccc}\text{Res}:\text{Sheaves}\left(X\right)& \hookrightarrow & \text{Presheaves}\left(X\right)\\ 𝒢& \mapsto & 𝒢\end{array}$

Sheafification is the left adjoint functor to inclusion $\text{Sh:}\text{Presheaves}\left(X\right)\to \text{Sheaves}\left(X\right)$ given by

${\text{Hom}}_{\text{Sheaves}\left(X\right)}(\text{Sh}\left(ℱ\right),𝒢)\simeq {\text{Hom}}_{\text{Presheaves}\left(X\right)}(ℱ,\text{Res}\left(𝒢\right))$

HW 5: Show that the sheafification of a presheaf $ℱ$ is given by

see Macdonald.

HW 6: Show that

1. $\text{Sheaves}\left(X\right)$ is a subcategory of $\text{Presheaves}\left(X\right)$
2. $\text{Sheaves}\left(X\right)$ is an abelian category
3. $\text{Presheaves}\left(X\right)$ is an abelian category
4. $\text{Sheaves}\left(X\right)$ is not an abelian subcategory of $\text{Presheaves}\left(X\right)$

HW 7:

Let $X={ℂ}^{\times }$.

Define sheaves $𝒪,{𝒪}^{\times }$ and $\mathbb{Z}$ on $X$ by

$\begin{array}{ccc}𝒪\left(U\right)& =& \{\text{continuous}f:U\to ℂ\}\\ {𝒪}^{\times }\left(U\right)& =& \{\text{continuous}f:U\to {ℂ}^{\times }\}\\ \mathbb{Z}\left(U\right)& =& \{\text{continuous}f:U\to \mathbb{Z}\}\text{,}\end{array}$

for $U$ open in ${ℂ}^{\times }$. Thus

$\begin{array}{ccc}𝒪\left(X\right)& =& \{\text{continuous}f:{ℂ}^{\times }\to ℂ\}\\ {𝒪}^{\times }\left(X\right)& =& \{\text{continuous}f:{ℂ}^{\times }\to {ℂ}^{\times }\}\\ \mathbb{Z}\left(X\right)& =& \{\text{continuous}f:{ℂ}^{\times }\to \mathbb{Z}\}\text{,}\end{array}$

Show that

$0\to \mathbb{Z}\stackrel{2\pi i}{\to }𝒪\stackrel{\text{exp}}{\to }{𝒪}^{\times }\to 0$

is an exact sequence of sheaves on ${ℂ}^{\times }$ which is not an exact sequence of presheaves on ${ℂ}^{\times }$.

Show that

$\text{coKer}\left(𝒪\stackrel{\text{exp}}{\to }{𝒪}^{\times }\right)=\{\begin{array}{ccc}0,& & \text{in Sheaves}\left(X\right),\\ \mathbb{Z}=H^{\prime }(X,\mathbb{Z}),& & \text{in Presheaves}\left(X\right),\end{array}$

and $\text{coKer}\left(𝒪\stackrel{\text{exp}}{\to }{𝒪}^{\times }\right)$ in $\text{Presheaves}\left(X\right)$ is generated by the function

$\begin{array}{ccc}f:{ℂ}^{\times }& \to & {ℂ}^{\times }\\ z& \mapsto & \frac{1}{z}\end{array}$

which is a function in ${𝒪}^{\times }$ not in the image of $𝒪$ in $\text{Presheaves}\left(X\right)$.

Notes and References

The section on sheaves and ringed spaces follows the presentations in [Go, §1.9], [Mac, ???] and [Bo, §AG4.2]. It is common to define a presheaf as a (contravariant) functor $𝒯\to 𝒜$. Historically this was done because the language of categories was unfamiliar to most working research mathematicians, but this is no longer the case.

The section on direct and inverse images is based on the exposition of sheaves and presheaves in [Weibel]: the definitions of $\text{Presheaves}\left(X\right)$ and $\text{Sheaves}\left(X\right)$ are in [Weibel, Def. 1.6.5], HW 2 is [Weibel, Ex. 1.6.2], HW 7 is between [Weibel, Ex. 1.6.2] and [Weibel, Def. 1.6.6], the definition of sheafification is in [Weibel, Example 1.6.7], HW 3 and 4 are [Weibel, Example 2.3.2 and Exercise 2.3.6], the global sections functor is defined in [Weibel, Application 2.5.4] and HW 1 is in [Weibel, Exercise 2.6.3] and the direct image and inverse image are in [Weibel, Application 2.6.6].

The section on Presheaves and Sheafification is also based on the treatment in [Weibel].

References

[Bo] A. Borel, Linear Algebraic Groups, Section AG4.2, Graduate Texts in Mathematics 126, Springer-Verlag, Berlin, 1991, MR??????

[Go] R. Godement, Topologie algébrique et théorie des faisceaux, Section 1.9, Actualités scientifiques et industrielles 1252, Hermann, Paris, 1958. MR??????

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