Properties of The Stalk
As outlined in the introduction, stalks capture the local behaviour of a sheaf. As a sheaf is supposed to be determined by its local restrictions (see gluing axiom), it can be expected that the stalks capture a fair amount of the information that the sheaf is encoding. This is indeed true:
- A morphism of sheaves is an isomorphism, epimorphism, or monomorphism, respectively, if and only if the induced morphisms on all stalks have the same property. (However it is not true that two sheaves, all of whose stalks are isomorphic, are isomorphic, too, because there may be no map between the sheaves in question).
In particular:
- A sheaf is zero (if we are dealing with sheafs of groups), if and only if all stalks of the sheaf vanish. Therefore the exactness of a given functor can be tested on the stalks, which is often easier as one can pass to smaller and smaller neighbourhoods.
Both statements are false for presheaves. However, stalks of sheaves and presheaves are tightly linked:
- Given a presheaf P and its sheafification F, the stalks of P and F agree.
Read more about this topic: Stalk (sheaf)
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