Ringed Spaces

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 18 November 2009

Ringed Spaces

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

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

A sheaf on $X$ is a functor $ℱ:𝒯\to 𝒜$ such that if $U\in 𝒯$ and $𝒮\subseteq 𝒯$ is an open cover of $U$ then $$ℱ\left(U\right)=\left\{\left(s_{\alpha }\right)\in \prod _{U_{\alpha }\in 𝒮}ℱ\left(U_{\alpha }\right)|{\varphi }_{\alpha }^{\alpha \beta }\left(s_{\alpha }\right)={\varphi }_{\beta }^{\alpha \beta }\left(s_{\beta }\right)\right\}$$ where $${\varphi }_{\alpha }^{\alpha \beta }:ℱ\left(U_{\alpha }\right)\to ℱ\left(U_{\alpha }\cap U_{\beta }\right)\text{for}{U}_{\alpha },U_{\beta }\in 𝒮$$ is $ℱ\left(U_{\alpha }\supseteq U_{\alpha }\cap U_{\beta }\right)$.

A morphism of sheaves is a morphism of functors.

A ringed space is a pair $\left(X,{𝒪}_X\right)$ where $X$ is a topological space and ${𝒪}_X$ is a sheaf of rings on $X$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)