Stalks

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: 16 November 2009

Stalks

Let X be a topological space, let x∈X and let ℱ:𝒯→𝒜 be a functor.

Let 𝒩X be the neighourhood filter at x.

The stalk of 𝒰 at x is ℱx=limU∈𝒩xℱU(direct limit).

Let E=∏x∈Xℱx with topology given by setting W open if it satisfies the following constraint. IfU∈𝒯ands∈ℱUthens~-1Wis open, where s~:U⟶Ex⟼sxis given byℱU⟶ℱxs⟼sx is the natural homomorphism.

The sheafification of ℱ is the sheaf ℱ~ given by ℱ~U=s′=s′x∈∏x∈Uℱx|ifx∈Uthen there existsV∈𝒯, ands∈ℱVsuch thatx∈V⊆U, ands′yy∈V=syy∈V=s′:U→E|s′is continuous andp∘s~=idU where p:E→X sends ℱx to x.

We have homomorphisms ℱU↪ℱ~Us↦sxx∈U.

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)