Local Rings – Comparing First-Order Theories of Complex Variety

Question

Let $X$ be a complex variety containing some point $x$. Then $X$ is naturally a complex-analytic space, and we have an inclusion of rings $\mathbb{C}[X]_x\hookrightarrow\mathbb{C}\{X\}_x\hookrightarrow\widehat{\mathbb{C}[X]_x}$; that is, the ring of holomorphic germs is sandwiched between the ring of polynomial germs and the ring of formal power series at $x$. A number of important algebraic properties of formal power series can also be shown to hold for holomorphic germs; one useful method of proof is to do a construction in $\widehat{\mathbb{C}[X]_x}$, then show that it doesn't leave $\mathbb{C}\{X\}_x$. The most fundamental of these results is probably the Weierstrass Preparation Theorem, which can be proven for holomorphic germs in this way (see Zariski-Samuel vol II, Ch 7.1). Another important such result is Hensel's lemma, which can again be proven algebraically as well as by using the implicit function theorem. (So the ring of holomorphic germs is actually sandwiched between the Henselization of the regular germs and the formal power series.)

I've been trying to find an appropriate way to describe this relationship, and I was wondering if it can be done with model theory. Of the four rings (polynomials, algebraic power series, holomorphic germs, power series) describing the local behavior of the variety, what is the relationship between their first-order theories (in the language of $\mathbb{C}$-algebras)? Are any of the inclusions elementary—and, if so, how should this be interpreted geometrically? Which of these substructures are definable? Is model theory even the right way to approach this question?

Answer 1

It turns out that all of these except the ring of polynomial germs are elementary equivalent. The following answer is from Elliot Kaplan (posted with permission):

I think it's hard to say in general what an elementary extension of commutative rings looks like, but a necessary condition is certainly that the smaller ring R is existentially closed in S (that is, any system of polynomial equations and inequations over R which has a solution in S also has a solution in R itself). Hans Schoutens has done a lot of work on the intersection of model theory and commutative algebra, and you may be able to find better answers in some of his papers: http://websupport1.citytech.cuny.edu/Faculty/hschoutens/index.html

Here's a specific instance where we know an extension to be elementary: suppose that R and S are both henselian valuation rings with S a local ring extension of R. Suppose also that R and S have the same residue field, that this residue field has characteristic zero, and that the fraction fields of R and S have the same value group (that is Frac(R)/U(R) = Frac(S)/U(S), where U(R) is the multiplicative group of units of R). Then the Frac(R) is an elementary extension of Frac(S) as valued fields, and so R is an elementary extension of S as commutative rings (this is an instance of the Ax-Kochen-Ershov theorem for henselian valued fields of residue characteristic zero)

This gives a positive answer when your ring of holomorphic germs is isomorphic to the ring of convergent power series over C in one variable, since this ring is henselian. In more than one variable, I don't know whether the extension is even existentially closed (I have a gut feeling that the extension is not elementary, but I don't have a good reason why not)

The ring of algebraic power series, convergent power series, and formal power series are all henselian valued fields and, as a corollary of Weierstrass Preparation, have the same value group. Consequently, these extensions are in fact elementary.