Reference for "It is enough to specify a sheaf on a basis"?

Question

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?

Answer 1

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"

Answer 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.

Answer 3

I'd like to add my own reference and strategy: Siegfried Bosch's Algebraic Geometry and Commutative Algebra has a pretty developped discussion on constructing $\mathcal{O}_X$ the structural sheaf of a scheme $X$ (cf. theorem 3 and Lemma 4 of section 6.6, p.242). However there is only a proof for a basis $\mathcal{B}$ stable under intersection, so I struggled for a while with this. It is only now that I fully grasp the ideas behind the Stacks Project's page on the matter. I'd like to show how to use it to its fullest here.

So let's fix the notation once more:

• $X$ is a topological space and $\mathcal B$ is a basis for its topology.
• For any open $U\in\mathcal B,~\,\rm{Cov}_{\mathcal B}(U)$, is the set of coverings $\mathcal{U}=(U_i)_{i\in I}$ of $U$, by opens from the basis $U_i\in\mathcal B$.
• For all opens $U\in\mathcal B$, set $C(U)\subset\rm{Cov}_{\mathcal B}(U)$ a cofinal system of coverings for the refinement relationship: for any covering $\mathcal U=(U_i)_{i\in I}\in\rm{Cov}_{\mathcal B}(U)$ there is a covering $\mathcal V=(V_j)_{j\in J}$ in $C(U)$ that refines $\mathcal U$, i.e., there is a map $\alpha:J\longrightarrow I$ such that $\forall j\in J,\,V_j\subseteq U_{\alpha(j)}$.
• For every $\mathcal U=(U_i)_{i\in I}\in C(U)$ and every $i,i'\in I$ fix once and for all a covering $\mathcal U_{i,i'}=(U_{i,i',k})_{k\in I_{i,i'}}$ of the intersection $U_i\cap U_{i'}$.

We now recall the statement oncerning sheaves on bases and refinements:

Let $\mathcal F$ be a given presheaf on $\mathcal B$. Then $\mathcal F$ is a $\mathcal B$-sheaf iff for any $U\in\mathcal B$ and any $\mathcal U=(U_i)_{i\in I}\in C(U)$, the following holds:

$(**)$For any collection of sections $s_i\in\mathcal F(U_i),i\in I$ such that $\forall i,i'\in I$ $$s_i|_{U_{i,i',k}} = s_{i'}|_{U_{i,i',k}}$$ there exists a unique section $s \in \mathcal{F}(U)$ such that $s_ i = s|_{U_ i}$ for all $i\in I$.

So there's a lot to unpack in this, but the main idea I want to point out is that we're almost done. If we can manage to define a pre-sheaf $\overline{\mathcal F}$ that extends a $\mathcal B$-sheaf $\mathcal F$, then we are done. Indeed we can choose our set of opens $\rm{O}(X)$ of $X$ to form a basis for its own topology, furthermore we define, for any open $U\subset X$ (!), $$C(U):=\rm{Cov}_\mathcal B(U)\subset\rm{Cov}_{\rm{O}(X)}(U)$$ and fix any coverings of the overlaps $\mathcal U_{i,i'}$ (which can be chosen to be reduced to a simple open set if the basis is stable under intersection). To prove cofinality, we just need to use the fact that any open is covered by opens in $\mathcal B$: take any covering $(U_i)_i$ of $U$ and cover every $U_i$ by opens in $\mathcal B$. By applying the above result we see that $\overline{\mathcal F}$ is ncessarily a sheaf.

To find a presheaf extension $\overline{\mathcal F}$ we can just use the classic Hartshorne definition using the espace étalé, but I prefer to use the more formal construction: $$\overline{\mathcal F}(U):=\lim_{V\in\mathcal B,V\subset U}\mathcal F(U).$$ Indeed, for any $U\in\mathcal B$ the projection maps $$\overline{\mathcal F}(U)\longrightarrow\mathcal F(U)$$ form a natural isomorphism.