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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?

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    @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

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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"

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    Thank you, this is very helpful!2011-11-28
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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.

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    @Rasmus Ok, fair enough.2011-11-28
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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.