Lingering foundational question about sheaves of abelian groups

Motivation for the question:

I have a standard working knowledge of sheaves. Given a scheme, a coherent module over its structure sheaf and a few hours I can compute things. Despite this I have always been unsure how to prove that the category of sheaves has all the properties which I know about.

Whenever I try and prove foundational results about sheaves, I end up producing circular arguments. The reason this happens is because for me, it is impossible to think about sheaves without my intuition kicking in. This is great for geometry, but not so great when I am trying to explain why you can check exactness at a local or stalk level to someone who is just starting Hartshorne. I always end up saying "it's just true!" or "its a standard exercise" which is never a good thing in mathematics.

I suspect the issue is that I don't really know how to prove that sheaves of abelian groups form an abelian category (at least i got stuck trying to do this).

Question: I have always struggled proving that sheaves of abelian groups form an abelian category. Is there a slick way to do this? I am not afraid of category theory.

• Proving that something is abelian involves quite a few verifications... What particular part do you have problems with? – Mariano Suárez-Álvarez Aug 30 '13 at 17:57
• Can't you realize sheaves of abelian groups as a quotient category of the category of presheaves by the Serre subcategory of presheaves that are zero locally, but not globally? – arsmath Aug 30 '13 at 18:48
• @arsmath Indeed, you can. Sheaves of abelian groups are the quotient of the presheaves by the kernel of the sheafification-functor. This Serre subcategory is localizing, from which one can conclude, that the sheaves are a even a Grothendieck category. Especially they must have injective envelopes, are bicomplete and can be realized (via Gabriel-Popescu theorem) as a quotient of a category of modules over a ring by a localizing subcategory. – archipelago Aug 30 '13 at 22:01
• Just do all the exercises in Chapter II Section 1 of Hartshorne (in full detail, working directly from the definition of a sheaf). – John Pardon Aug 31 '13 at 22:25
• As a reference for the category theory involved in abelian sheaves - in a direct, fundamental treatment - you could also take a look at "Categories and Sheaves" by Kashiwara and Schapira. The latter is more category theoretic and might suit you better: There is an explicit chapter on sheaves of abelian groups. – Gerrit Begher Sep 1 '13 at 18:13

2 Answers

I take it you mean sheaves on topological spaces. I think it is valuable to grasp both of two approaches. One (which works essentially the same way for sheaves on sites) is to first see that presheaves of Abelian groups have biproducts, kernels, and cokernels trivially by naturality; and then see by adjointness that sheafification of presheaf coproducts and cokernels gives coproducts and cokernels of sheaves of groups.

The other way is to think of sheaves on a topological space as local homeomorphisms from espaces etales, verify that products and kernels are defined very simply and that products also have the coproduct property (inherited from Abelian groups pointwise) and that cokernels are defined by a relatively simple local criterion.

Maybe I am missing what your problem is, though. Can you point to a part of these steps that gives you trouble?

• One also has to verify that the canonical homomorphism of sheaves $\mathrm{coker}(\ker(f)) \to \ker(\mathrm{coker}(f))$ is an isomorphism. One proof just reduces it, via stalks, to the case of abelian groups. I think that this cannot be formally reduced to the case of presheaves? – Martin Brandenburg Aug 31 '13 at 19:27
• @MartinBrandenburg I'm not sure what you have in mind. But your answer at mathoverflow.net/questions/89568/… notes sheafification is exact. Of course you have to prove that somehow, depending on how you approach sheafifiction. – Colin McLarty Aug 31 '13 at 23:12

Ok, I know I should have done this before I asked the question, but I had a dig through the stacks project. It seems like http://stacks.math.columbia.edu/tag/03A3 is exactly what I am looking for.