2
$\begingroup$

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?

Question also asked on MathOverflow

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, $\lim_{D(f)\in\mathcal{B}}F(D(f))=\mathcal{O}$.

$\endgroup$
2
  • 2
    $\begingroup$ Maybe is this usefull? $\endgroup$ Feb 9 at 12:19
  • $\begingroup$ @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. $\endgroup$
    – Anonymous
    Feb 10 at 4:20

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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 $\mathcal{O}$. $\endgroup$
    – Anonymous
    Feb 18 at 23:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.