# Is the structure presheaf of $\text{Spec}(A)$ a limit of presheaves?

In Shafarevich's Basic Algebraic Geometry, he defines the structure presheaf on $$\text{Spec}(A)$$ first by $$\mathcal{O}(D(f))\cong A_f$$ and then $$\mathcal{O}(U)=\lim_{D(f)\subset U}\mathcal{O}(D(f))$$, where $$f$$ is an element of the ring $$A$$, $$D(f)$$ is a Zariski basic open set, $$A_f$$ is the localization of $$A$$ at the multiplicative set $$\{f^n:n\in\mathbb{N}\}$$, and $$U$$ is an arbitrary Zariski open set. Given $$D(f)\subset D(g)$$, he defines explicitly the map $$\rho_{D(f)}^{D(g)}\mathcal{O}(D(g))\to\mathcal{O}(D(f))$$ but put succinctly, it is the unique map $$A_g\to A_f$$ induced by the natural map $$A\to A_f$$ using the universal property of $$A_g$$.

Under this definition, we get the universal property for $$\mathcal{O}(U)$$ that if $$R$$ is any ring together with a collection of homomorphisms $$\phi_f:R\to A_f$$ for each $$D(f)\subset U$$ such that these maps commute with the homomorphisms $$\rho_{D(f)}^{D(g)}$$, then there exists a unique homomorphism $$u:R\to\mathcal{O}(U)$$ such that $$\phi_f=\rho_{D(f)}^U\circ u$$ for all $$D(f)\subset U$$.

However, this implies a universal property about $$\mathcal{O}$$ itself: If $$F$$ is any presheaf of $$\text{Spec}(A)$$ with restriction maps $$res_V^U:F(U)\to F(V)$$, such that $$F(D(f))$$ is naturally isomorphic to $$A_f$$ for all $$f$$, then there exists a unique natural transformation $$\eta:F\to\mathcal{O}$$ such that $$res_{D(f)}^U=\rho_{D(f)}^U\circ\eta_U$$ for any $$D(f)\subset U$$ (modulo the natural isomorphism $$F(D(f))\cong A_f\cong\mathcal{O}(D(f))$$). This suggests to me that there should be a way to write $$\mathcal{O}$$ as a limit of presheaves, but it is not clear to me how to do that. So my question is, is it possible to express $$\mathcal{O}$$ as a limit of presheaves (in a nontrivial way) and if so, how?

Edit: I think I came up with a way to translate the universal property into limits. Specifically, $$\mathcal{O}$$ is the terminal object in a certain category.

Let $$\mathcal{B}$$ be the poset of basic open sets of $$\operatorname{Spec}(A)$$, and let $$\mathbf{Psh}(X)$$ denote the category of presheaves on a category. Let $$r:\mathbf{Psh}(\operatorname{Spec}(A))\to\mathbf{Psh}(\mathcal{B})$$ be the functor which restricts a presheaf on $$\operatorname{Spec}(A)$$ to the basic open sets, and let $$\mathcal{O}_{basic}$$ be the presheaf on $$\mathcal{B}$$ which sends $$D(f)$$ to $$A_f$$ (i.e. the structure sheaf only on basic open sets). Lastly, let $$\mathbf{C}$$ be the full subcategory of the comma category $$r/\mathcal{O}_{basic}$$ for which the morphisms to $$\mathcal{O}_{basic}$$ are isomorphisms (i.e. the objects of $$\mathbf{C}$$ are pairs $$(\mathcal{F},\phi)$$ where $$\mathcal{F}$$ is a presheaf on $$\operatorname{Spec}(A)$$ and $$\phi$$ is a natural isomorphism from $$r(\mathcal{F})$$ to $$\mathcal{O}_{basic}$$). Then $$\mathcal{O}$$ is the terminal object of $$\mathbf{C}$$. I don't know if there is a better way to express the universal property I stated as a limit than this, but I would be happy to hear it if there is.

There are two other ways to express $$\mathcal{O}$$ as a limit of presheaves or sheaves which I now know (neither or which is a direct translation of the universal property I stated). The first is to use the left adjoint to the restriction $$\mathbf{Sh}(\operatorname{Spec}(A))\to\mathbf{Sh}(\mathcal{B})$$ as explained in John's answer. The other way is to take Shafarevich's limit definition of $$\mathcal{O}(U)$$ as a limit and use the fact that a limit of presheaves is computed component-wise. Explicitly, let $$F:\mathcal{B}\to\mathbf{Psh}(\operatorname{Spec}(A))$$ be the functor which takes an open set $$D(f)$$ and sends it to the presheaf given by $$F(D(f))(U)=A_f$$ if $$D(f)\subseteq U$$ and $$F(D(f))(U)=0$$ otherwise. The restriction maps are all either the identity map or the zero map. Then $$\left(\lim_{D(f)\in\mathcal{B}} F(D(f))\right)(U)=\lim_{D(f)\in\mathcal{B}}F(D(f))(U)=\lim_{D(f)\subseteq U}A_f=\mathcal{O}(U)$$ and thus, $$\lim_{D(f)\in\mathcal{B}}F(D(f))=\mathcal{O}$$.

• Maybe is this usefull? Feb 9 at 12:19
• @Armandoj18eos Thanks, but that's not quite what I was looking for. Your answer describes $\mathcal{O}(U)$ as a limit of the $\mathcal{O}(D(f))$, but I was looking an expression of $\mathcal{O}$ as a limit of presheaves of some sort. John's answers looks like what I am looking for, so I will likely accept his answer when I get a chance to read it in detail. Feb 10 at 4:20

One way to capture your intuition here, I think, is that $$\mathscr{O}$$ is somehow "fully determined" by its values on distinguished affine subsets $$D(f) \subseteq \operatorname{Spec} A$$. One way to make this intuition precise is to show that the category of sheaves on the Zariski site of $$\operatorname{Spec} A$$ is equivalent to the category of sheaves on distinguished affine site of $$\operatorname{Spec} A$$. This equivalence is discussed in $$\S13.3$$ of Vakil's Foundations of Algebraic Geometry, for example. (Don't be put off by the chapter number! The prerequisites for this subsection are really just the definition of sheaves on a scheme.) In particular, this captures your observation that if $$F$$ is a presheaf on the Zariski topology that's naturally isomorphic to $$\mathscr{O}$$ on the distinguished affine base, then $$F$$ and $$\mathscr{O}$$ are isomorphic as (pre)sheaves on the Zariski topology.
Write $$\operatorname{Sh}(Z)$$ for the category of sheaves on the Zariski site and $$\operatorname{Sh}(D)$$ for the category of sheaves on the distinguished affine site. The equivalence above gives a restriction map $$r^{*} : \operatorname{Sh}(Z) \to \operatorname{Sh}(D)$$; this is just the map sending a sheaf $$F$$ on the Zariski topology to the sheaf $$r^{*}F$$ given by $$(r^{*}F)(D(f)) = F(D(f))$$. In particular, we have $$(r^{*}\mathscr{O})(D(f)) = A_{f}$$. Since $$r^{*}$$ is an equivalence, it has a left adjoint $$L : \operatorname{Sh}(D) \to \operatorname{Sh}(Z)$$, and left adjoints can always be computed by limits: we have $$LF = \lim_{F \to (r^{*}G)} G$$ more or less by definition of a left adjoint. So you can write $$\mathscr{O}$$ as $$Lr^{*}\mathscr{O}$$, expressing it as a limit. Perhaps this limit is nontrivial.
In fact there is another way to express sheaves on the Zariski site as limits. The restriction map $$r^{*} : \operatorname{Sh}(Z) \to \operatorname{Sh}(D)$$ is induced by the inclusion $$r : D \to Z$$ of the distinguished affine site into the Zariski site. Since $$D$$ is a dense subsite of $$Z$$ (more or less by definition of $$Z$$), the right adjoint $$r_{*} : \operatorname{Sh}(D) \to \operatorname{Sh}(Z)$$ is the right Kan extension, and right Kan extensions can be computed by limits. So, for example, for any $$U \subseteq \operatorname{Spec} A$$ we have $$(r_{*}F)(U) = \lim_{D(f) \subseteq U} F(D(f))$$ In the case of $$F = \mathscr{O}$$ you recover the definition you started off with.
So I think that the nontrivial thing here is that the distinguished affine site $$D$$ is a dense sub-site of the Zariski site $$Z$$; this leads to the equivalence of $$\operatorname{Sh}(D)$$ and $$\operatorname{Sh}(Z)$$ in the particularly simple form that Shafarevich uses to define the structure presheaf on $$\operatorname{Spec} A$$. Note that none of this depends on the fact that $$X = \operatorname{Spec} A$$ is affine, and so the structure sheaf of any scheme $$X$$ is the sheaf in $$\operatorname{Sh}(Z)$$ corresponding to the tautological sheaf $$F$$ acting as $$F(\operatorname{Spec} A) = A$$ on the distinguished affine site. Of course, usually one defines the distinguished affine site of a scheme $$X$$ by reference to the structure sheaf $$\mathscr{O}_{X}$$, so this really is pretty tautological.
• Thanks, this is great! The limit you describe is wonderful, though it doesn't directly translate the universal property I wrote from what I can tell (sorry if it wasn't clear this was desired). If I am reading right, the universal property for that limit is that for any sheaf $F$ of $Z$ with a natural transformation from the basic structure sheaf $\mathcal{O}_{basic}\to r*F$, there is a natural transformation $\mathcal{O}\to F$ whose image under $r*$ commutes with the transformations $\mathcal{O}_{basic}\to r*F$, and any other sheaf which does this has a unique transformation to $\mathcal{O}$. Feb 18 at 23:46