# Is the sheaf of rings $\mathscr{O}$ the sheafification of a presheaf?

This a paragraph from Hartshorne's Algebraic Geometry:

Next we will define a sheaf of rings $$\mathscr{O}$$ on $$\text{Spec }A$$. For each prime ideal $$\mathfrak{p}\subseteq A$$, let $$A_{\mathfrak{p}}$$ be the localization of $$A$$ at $$\mathfrak{p}$$. For an open set $$U\subseteq \text{Spec }A$$, we define $$\mathscr{O}(U)$$ to be the set of functions $$s:U\to \coprod_{\mathfrak{p}\in U}A_{\mathfrak{p}}$$, such that $$s(\mathfrak{p})\in A_{\mathfrak{p}}$$ for each $$\mathfrak{p}$$, and such that $$s$$ is locally a quotient of elements of $$A$$: to be precise, we require that for each $$\mathfrak{p}\in U$$, there is a neighborhood $$V$$ of $$\mathfrak{p}$$, contained in $$U$$, and elements $$a, f\in A$$, such that for each $$\mathfrak{q}\in V$$, $$f\notin \mathfrak{q}$$, and $$s(\mathfrak{q})=a/f$$ in $$A_{\mathfrak{q}}$$. (Note the the similarity with the definition of the regular functions on a variety. The difference is that we consider functions into the various local rings, instead of to a field.)

My Question: $$\mathscr{O}$$ seems to be the sheafification of a presheaf. Is it? What is it?

I have compared the construction of the sheafification of a presheaf:

We construct the sheaf $$\mathscr{F}^+$$ as follows. For any open set $$U$$, let $$\mathscr{F}^+(U)$$ be the set of functions $$s$$ from $$U$$ to the union $$\bigcup_{P\in U}\mathscr{F}_P$$ of the stalks of $$\mathscr{F}$$ over points of $$U$$, such that

(1) for each $$P\in U$$, $$s(P)\in \mathscr{F}_P$$, and

(2) for each $$P\in U$$, there is a neighborhood $$V$$ of $$P$$, contained in $$U$$, and an element $$t\in \mathscr{F}(V)$$, such that for all $$Q\in V$$, the germ $$t_Q$$ of $$t$$ at $$Q$$ is equal to $$s(Q)$$.

I think it should have $$\varinjlim_{\mathfrak{p}\in U}\mathscr{F}(U)=\mathscr{F}_{\mathfrak{p}}=A_{\mathfrak{p}}=\varinjlim_{f\in A\backslash \mathfrak{p}}A_f.$$ But I don't know how to defin $$\mathscr{F}(U)$$...

Edit. I know that every sheaf is a presheaf itself. However, we have not proven that $$\mathscr{O}$$ is a sheaf. It is the task of the following paragraph in the book.

Now it is clear that sums and products of such funtions are again such, and that the element $$1$$ which gives $$1$$ in each $$A_{\mathfrak{p}}$$ is an identity. Thus $$\mathscr{O}(U)$$ is a commutative ring with identity. If $$V\subseteq U$$ are two open sets, the natural restriction map $$\mathscr{O}(U)\to \mathscr{O}(V)$$ is a homomorphism of rings. It is then clear that $$\mathscr{O}$$ is a presheaf. Finally, it is clear from the local nature of the definition that $$\mathscr{O}$$ is a sheaf.

That is why I am looking for a presheaf which induces $$\mathscr{O}$$.

• Every sheaf is a presheaf, and is the sheafification of itself. – Lord Shark the Unknown Apr 12 at 16:45
• – André 3000 Apr 12 at 17:25