Topics in Abstract Algebra/Sheaf theory

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Sheaf theoryEdit

We say   is a pre-sheaf on a topological space   if

  • (i)   is an abelian group for every open subset  
  • (ii) For each inclusion  , we have the group morphism   such that
      is the identity and   for any inclusion  

A pre-sheaf is called a sheaf if the following "gluing axiom" holds:

For each open subset   and its open cover  , if   are such that   in  , then there exists a unique   such that   for all  .

Note that the uniqueness implies that if   and   for all  , then  . In particular,   for all   implies  .

4 Example: Let   be a topological group (e.g.,  ). Let   be the set of all continuous maps from open subsets   to  . Then   forms a sheaf. In particular, suppose the topology for   is discrete. Then   is called a constant sheaf.

Given sheaves   and  , a sheaf morphism   is a collection of group morphisms   satisfying: for every open subset  ,


where the first   is one that comes with   and the second  .

Define   for each open subset  .   is then a sheaf. In fact, suppose  . Then there is   such that  . But since


for all  , we have  . Unfortunately,   does not turn out to be a sheaf if it is defined in the same way. We thus define   to be the set of all   such that there is an open cover   of   such that   is in the image of  . This is a sheaf. In fact, as before, let   be such that  . Then we have an open cover of   such that   restricted to each member   of the cover is in the image of  .

Let   be sheaves on the same topological space.

A sheaf   on   is said to be flabby if   is surjective. Let  , and, for each  , define  .   is closed since   implies   has a neighborhood of   such that   for every  . Define  . In particular, if   is a closed subset and  , then the natural map   is an isomorphism.

4 Theorem Suppose


is exact. Then, for every open subset  


is exact. Furthermore,   is surjective if   is flabby.
Proof: That the kernel of   is trivial means that   has trivial kernel for any  . Thus the first map is clear. Next, denoting   by  , suppose   with  . Then there exists an open cover   of   and   such that  . Since   in   and   is injective by the early part of the proof, we have   in   and so we get   such that  . Finally, to show that the last map is surjective, let  , and  . If   is totally ordered, then let  . Since   agree on overlaps by totally ordered-ness, there is   with  . Thus,   is an upper bound of the collection  . By Zorn's Lemma, we then find a maximal element  . We claim  . Suppose not. Then there exists   with  . Since   in  , by the early part of the proof, there exists   with  . Then   (so  ) while   in  . This contradicts the maximality of  . Hence, we conclude   and so  .  

4 Corollary


is exact if and only if


is exact for every  .

Suppose   is a continuous map. The sheaf   (called the pushforward of   by  ) is defined by   for an open subset  . Suppose   is a continuous map. The sheaf   is then defined by   the sheafification of the presheaf   where   is an open subset of  . The two are related in the following way. Let   be an open subset. Then   consists of elements   in   where  . Since  , we find a map


by sending   to  . The map is well-defined for it doesn't depend on the choice of  .