# When do the spectra of overrings glue to a proper morphism?

This question is motivated by the construction of blowups.

Let $$A \subset K$$ be a commutative domain and its fraction field, and let $$\{A_i\}$$ be some finite collection of overrings in between.

Let $$X = \mathrm{Spec}A$$ and let $$\tilde{X} = \bigcup U_i$$ be the $$X$$-scheme formed by gluing the overing spectra via inclusions of (generated) subrings.

So $$\mathrm{Spec}(A_i) \cap \mathrm{Spec}(A_j) = \mathrm{Spec}(A_i+A_j)$$, and so on.

If an overring is a localization, then its induced morphism to $$X$$ is an open immersion, but this need not be the case. However, I believe it is the case that the ideal class group of $$A$$ is trivial if and only if each such overring is a localization.

For example, if $$A = k[x,y], A_0 = k[x,y,\frac{x}{y}], A_1 = k[x.y.\frac{y}{x}], A_{01} = k[x^{\pm 1}, y^{\pm 1}]$$, then the the overrings glue to the blowup of the plane at the origin.

Question: When do the overrings glue to a proper morphism $$\tilde{X} \to X$$? What are some good criteria for properness, and how to check it?

Bonus question: When is the morphism finite?

In the blowup example, $$\tilde{X}$$ is proper but not finite, and upon removal of either overring from the collection, the remaining subscheme is not proper.

• @RizaHawkeye Yes, that's true. The reason I mentioned removing some of the $A_i$ from the collection is to give an example of subsets of the $A_i$ that provide different answers. Instead I could add in $A_2 = k[x,y,\frac{x}{y+1}], A_{02} = k[x,y,\frac{1}{x}, \frac{1}{y+1}]$ and obtain a scheme with more affine patches which is not proper. So I'm looking for nice criteria to identify which of $\{A_0, A_1, A_{01}\}, \{A_0, A_{01}\}, \{A_1, A_{01}\}, \{A_0, A_1, A_{01}, A_2, A_{02}\}$ define a proper morphism to $A$. – PrimeRibeyeDeal Jul 20 at 19:49