10
$\begingroup$

The wikipedia article on sheaves says:

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis.

However, it does not cite a specific reference for this statement. Does there exist a rigorous proof for this statement in the literature?

  • 1
    I would say this is a standard exercise which everyone should do at some point!2011-11-28
  • 2
    @Mariano: Suppose you want to use this fact in an article you write. Do you $$ $$ a) claim it without proof, $$ $$ b) prove it, or $$ $$ c) try to find a proper reference?2011-11-28
  • 0
    If you are writing a paper in which sheaves show up, you should be able to prove this, since it is really just manipulation of the definitions. I insist: this is an exercise with sheaves I expect anyone talking to me about the subject will be able to do.2011-11-28
  • 0
    @Mariano: What's so terrible about giving a decent reference for a standard fact instead of just skidding over the details?2011-11-28
  • 1
    There is nothing terrible, but this is a *basic* fact (as George observes, this is precisely how one defines the structural sheaf of an affine scheme or the sheaf associated to a module; this is Chapter 2 in Hartshorne!) This is something that everyone should do when she is learning this stuff; I really think that a reference to a proof of this particular fact would be pretty out of place in a research paper.2011-11-29
  • 2
    @Mariano: Let me make some explaining remarks: I'm not in algebraic geometry. What happened really was like this: So we have this complicated K-theory invariant for C*-algebras equipped with a system of distinguished ideals. Let's assume said algebra has real rank zero. Then this part of the invariant looks like a sheaf on a certain non-Hausdorff space. And that other part looks like a cosheaf. Aren't (co)sheafs determined by what they are doing on a basis? Then our invariant has a lot of redundance! So we would like to use a fact that should be well-known to algebraic geometers...2011-11-29
  • 2
    ... let's hope we can find a proper reference, because this is certainly not a triviality for our readers! $$ $$ By the way, this fact is basic enough for its dual version for cosheafs (covariant functors on open sets satisfying a condition you might call co-gluing) to hold, isn't it?2011-11-29
  • 0
    @Mariano Unfortunately (IMHO) this is not how Hartshorne describes the sheaf associated to a module (or the structure sheaf of an affine scheme). He defines it in terms of actual functions on open sets, and then proves that it gives what it's supposed to on the standard open subsets.2012-04-17

3 Answers 3

10

This is an excellent question and to tell the truth it is often handled in a cavalier fashion in the literature. This is a pity because it is a fundamental concept in algebraic geometry.

For example the structural sheaf $\mathcal O_X$ of an affine scheme $X=Spec(A)$ is defined by saying that over a basic open set $D(f)\subset X \;(f\in A)$ its value is $\Gamma(D(f),\mathcal O_X)=A_f$ and then relying on the mechanism of sheaves on a basis to extend this to a sheaf on $X$.
The same procedure is also followed in defining the quasi-coherent sheaf of modules $\tilde M$ on $X$ associated to the $A$-module $M$.

However there are happy exceptions on the net , like Lucien Szpiro's notes where sheaves on a basis of open sets are discussed in detail on pages 14-16.
You can also find a careful treatment in De Jong and collaborators' Stack Project , Chapter 6 "Sheaves on Spaces", section 30, "Bases and sheaves"

  • 0
    Thank you, this is very helpful!2011-11-28
2

It is given in Daniel Perrin's Algebraic Geometry, Chapter 3, Section 2. And by the way, it is a nice introductory text for algebraic geometry, which does not cover much scheme theory, but gives a definition of an abstract variety (using sheaves, like in Mumford's Red book).

Added: I just saw that Perrin leaves most of the details to the reader. For another proof, see Remark 2.6/Lemma 2.7 in Qing Liu's Algebraic Geometry and Arithmetic curves.

  • 1
    I was about to give the reference to Qing Liu's book too. I found it very useful.2011-11-28
  • 0
    Thanks for the references. However, Liu does not proof anything in Remark 2.6. And Lemma 2.7 is just a reformulation of the definition of a sheaf.2011-11-28
  • 0
    @Rasmus I don't think it is fair to say that Liu does not prove anything. However, I do agree that not all details are given, just as in Perrin, but using both point of view, can you fill in the details missing? That way, you will see of you have a good understanding of the basic axioms of a sheaf.2011-11-28
  • 2
    I think my understanding of the basic axioms of a sheaf is fine. What I am looking for is a reference I can cite, so that I do *not* have to reproduce these standard arguments.2011-11-28
  • 1
    @Rasmus Ok, fair enough.2011-11-28
1

This is proven in Serres FAC, Chapter 1 Section 1 subsection 4.

His definition of a sheaf is what is currently called an etale space and a modern pre sheaf is what Serre refers to as a system, then a modern sheaf is a system satisfying propositions 1 and 2. The categories of sheaves over X and etale spaces over X are equivalent though.

Edit: I originally said subsection 3, but it is subsection 4.