stalk


Let F be a presheafPlanetmathPlanetmathPlanetmath over a topological spaceMathworldPlanetmath X with values in an abelian categoryMathworldPlanetmathPlanetmathPlanetmath 𝒜, and suppose direct limitsMathworldPlanetmath exist in 𝒜. For any point p∈X, the stalk Fp of F at p is defined to be the object in 𝒜 which is the direct limit of the objects F⁢(U) over the directed set of all open sets U⊂X containing p, with respect to the restrictionPlanetmathPlanetmathPlanetmath morphismsMathworldPlanetmathPlanetmath of F. In other words,

Fp:=lim⟶U∋p⁢F⁢(U)

If 𝒜 is a category consisting of sets, the stalk Fp can be viewed as the set of all germs of sectionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of F at the point p. That is, the set Fp consists of all the equivalence classesMathworldPlanetmathPlanetmath of ordered pairsMathworldPlanetmath (U,s) where p∈U and s∈F⁢(U), under the equivalence relation (U,s)∼(V,t) if there exists a neighborhoodMathworldPlanetmathPlanetmath W⊂U∩V of p such that resU,W⁡s=resV,W⁡t.

By universal propertiesMathworldPlanetmath of direct limit, a morphism ϕ:F⟶G of presheaves over X induces a morphism ϕp:Fp⟶Gp on each stalk Fp of F. Stalks are most useful in the context of sheaves, since they encapsulate all of the local data of the sheaf at the point p (recall that sheaves are basically defined as presheaves which have the property of being completely characterized by their local behavior). Indeed, in many of the standard examples of sheaves that take values in rings (such as the sheaf 𝒟X of smooth functions, or the sheaf 𝒪X of regular functions), the ring Fp is a local ring, and much of geometry is devoted to the study of sheaves whose stalks are local rings (so-called “locally ringed spaces”).

We mention here a few illustrations of how stalks accurately reflect the local behavior of a sheaf; all of these are drawn from [1].

  • •

    A morphism of sheaves ϕ:F⟶G over X is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if and only if the induced morphism ϕp is an isomorphism on each stalk.

  • •

    A sequencePlanetmathPlanetmath F⟶G⟶H of morphisms of sheaves over X is an exact sequencePlanetmathPlanetmathPlanetmath at G if and only if the induced morphism Fp⟶Gp⟶Hp is exact at each stalk Gp.

  • •

    The sheafificationPlanetmathPlanetmath F′ of a presheaf F has stalk equal to Fp at every point p.

References

  • 1 Robin Hartshorne, Algebraic GeometryMathworldPlanetmathPlanetmath, Springer–Verlag New York Inc., 1977 (GTM 52).
Title stalk
Canonical name Stalk
Date of creation 2013-03-22 12:37:15
Last modified on 2013-03-22 12:37:15
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Definition
Classification msc 54B40
Classification msc 14F05
Classification msc 18F20
Related topic Sheaf
Related topic LocalRing