Let be a presheaf over a topological space with values in an abelian category , and suppose direct limits exist in . For any point , the stalk of at is defined to be the object in which is the direct limit of the objects over the directed set of all open sets containing , with respect to the restriction morphisms of . In other words,
If is a category consisting of sets, the stalk can be viewed as the set of all germs of sections of at the point . That is, the set consists of all the equivalence classes of ordered pairs where and , under the equivalence relation if there exists a neighborhood of such that .
By universal properties of direct limit, a morphism of presheaves over induces a morphism on each stalk of . Stalks are most useful in the context of sheaves, since they encapsulate all of the local data of the sheaf at the point (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 of smooth functions, or the sheaf of regular functions), the ring 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 .
The sheafification of a presheaf has stalk equal to at every point .
- 1 Robin Hartshorne, Algebraic Geometry, Springer–Verlag New York Inc., 1977 (GTM 52).
|Date of creation||2013-03-22 12:37:15|
|Last modified on||2013-03-22 12:37:15|
|Last modified by||djao (24)|