Sheaves and Ringed Spaces

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

Last updates: 01 January 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 $\mathrm{im}i=\ker (j-k)$.

Notes and References

These notes follow 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.

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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