$𝒪$-modules

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

$𝒪$-modules

Let $\left(X,𝒪\right)$ be a ringed space. An $𝒪$-module is a sheaf $ℱ$ of abelian groups on $X$ such that

1. if $U$ is open in $X$ then $ℱ\left(U\right)$ is an $𝒪\left(U\right)$-module

and the $𝒪\left(U\right)$-module structures are functional.

A sequence $ℱ\stackrel{\varphi }{\to }𝒢\stackrel{\psi }{\to }\mathscr{H}$ of $𝒪$-modules is exact if and only if ${ℱ}_x\stackrel{{\varphi }_x}{\to }{𝒢}_x\stackrel{{\psi }_x}{\to }{\mathscr{H}}_x$ is exact for all $x\in X$.

The global section function $\Gamma :\left\{𝒪\text{-modules}\right\}\to \left\{𝒪\left(X\right)\text{-modules}\right\}$ is defined by $$\Gamma \left(ℱ\right)=ℱ\left(X\right)$$ and the right derived functions of $\Gamma$ are $H^p\left(X,ℱ\right)=R^p\Gamma \left(𝒪\right)\text{.}$

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)