𝒪 -modules

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 18 November 2009

𝒪 -modules

Let X 𝒪 be a ringed space. An 𝒪 -module is a sheaf of abelian groups on X such that

  1. if U is open in X then U is an 𝒪 U -module
and the 𝒪 U -module structures are functional.

A sequence ϕ 𝒢 ψ of 𝒪 -modules is exact if and only if x ϕ x 𝒢 x ψ x x is exact for all x X .

The global section function Γ : 𝒪 -modules 𝒪 X -modules is defined by Γ = X and the right derived functions of Γ are H p X = R p Γ 𝒪 .

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)