# Can the structure sheaf of the spectrum of a ring be defined by taking appropriate localizations on every open set?

I recently learned that the structure sheaf on the spectrum of a ring $$\mathrm{Spec}(R)$$ is first defined on the distinguished open subsets $$D_f$$ for $$f\in R$$ so that $$\mathcal{O}(D_f)=R_f$$ where $$R_f$$ is the localization of the ring $$R$$ at $$f$$. Then the structure sheaf is extended to arbitrary open sets by taking limits. I was wondering if it was possible to define the structure sheaf more directly, by giving an explicit definition over an arbitrary open set. First I ran into the question Why is the structure sheaf for the spectrum of a ring defined locally? where the author tries to do something similar by defining, for an arbitrary open set $$U=\mathrm{Spec}(R)-V(I),$$ the ring $$\mathcal{O}(U)$$ to be the localization of $$R$$ at $$I$$. This definition fails because some of the functions $$g\in I$$ actually vanish in $$U$$, so that $$U\cap V(g)\neq \varnothing.$$ Then we end up allowing division by the function $$g$$ even though it is zero somewhere in $$U$$.

My idea, for arbitrary open $$U$$, was to define $$S=\{f\in R: U\cap V(f)=\varnothing\}.$$ Then $$S$$ is multiplicatively closed because if $$f,g\in S$$ then $$U\cap V(fg)=U\cap (V(f)\cup V(g))=\varnothing$$ so $$fg\in S.$$ Then we could define $$\mathcal{O}(U)=S^{-1}R.$$

Intuitively, this allows us to divide by any function that does not vanish over $$U,$$ and it agrees with the standard definition of the structure sheaf over the distinguished open sets. This is a bit different from the standard definition of the structure sheaf which only requires that a section over an open set $$U$$ does not divide by any function that vanishes "locally." If it gives a sheaf then it will actually be the same as the standard structure sheaf, since a sheaf will be uniquely determined by its sections over the distinguished open sets. I suspect that there may be some situations where this pre-sheaf fails gluability, but I can't think of any. Does this pre-sheaf fail the sheaf axioms in some cases?

• Mar 31 at 22:46

Let $$k$$ be a field, $$R=k[x,y,z,w]/(xy-zw)$$, and $$U=D_y\cup D_z$$. Note that the elements $$w/y\in R_y$$ and $$x/z\in R_z$$ are the same in $$R_{yz}$$ and so should glue to give an element of $$\mathcal{O}(U)$$. However, it can be shown that this element cannot be represented by a fraction whose denominator does not vanish on $$U$$. (Any denominator for this element has to be in the ideal $$(y,z)$$, but there is no single element of $$(y,z)$$ which vanishes only on $$V(y,z)$$.) So, your definition will fail the gluing axiom in this case.
• Thanks! Do you know if there are rings $R$ where my definition does give a sheaf? I was wondering if my definition might work for $R=k[x,y]$ or $R=C^\omega(\mathbb{C})$ for example. Aug 13, 2020 at 23:42
• Well, for instance, it works for rings in which every ideal is principal, since then every open set has the form $D_x$. I believe it should also work for any UFD, since then every fraction has a unique "minimal" denominator you can always use to glue. Aug 14, 2020 at 0:54