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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 [1].
- 1
- Robin Hartshorne, Algebraic Geometry, Springer-Verlag New York Inc., 1977 (GTM 52).
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"stalk" is owned by djao. [ full author list (2) ]
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(view preamble)
Cross-references: sheafification, exact sequence, sequence, induced, isomorphism, morphism of sheaves, reflect, geometry, local ring, regular functions, smooth functions, rings, property, sheaf, sheaves, induces, presheaves, universal properties, neighborhood, equivalence relation, ordered pairs, equivalence classes, sections, germs, category, morphisms, restriction, open sets, directed set, object, point, direct limits, abelian category, topological space, presheaf
There are 15 references to this entry.
This is version 6 of stalk, born on 2002-04-28, modified 2005-04-03.
Object id is 2881, canonical name is Stalk.
Accessed 4867 times total.
Classification:
| AMS MSC: | 18F20 (Category theory; homological algebra :: Categories and geometry :: Presheaves and sheaves) | | | 14F05 (Algebraic geometry :: homology theory :: Vector bundles, sheaves, related constructions) | | | 54B40 (General topology :: Basic constructions :: Presheaves and sheaves) |
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Pending Errata and Addenda
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