Timeline for Cardinalities larger than the continuum in areas besides set theory
Current License: CC BY-SA 2.5
7 events
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Nov 4, 2010 at 18:04 | comment | added | BCnrd | Dear Charles: These statements going from $\mathbf{C}$ to $\overline{\mathbf{F}}_p$ for sufficiently large $p$ are amenable to simple geometric proofs without model theory by using nothing much more than the Nullstellensatz and elementary flatness stuff, and such methods are extremely useful: for example, it's exactly what's needed to justify that various results at the geometric generic fiber in are equivalent to results on geometric fibers over a Zariski-dense open locus in the base (can then be applied over Spec($\mathbf{Z}$)!) | |
Nov 4, 2010 at 17:35 | comment | added | Charles Staats | BCnrd: I doubt that model-theoretic methods can provide proofs for this, since there can be first-order statements that are true for $R$ but not for any of the $R_i$, e.g., $\forall x \exists y (y^2=x)$. I would point out, however, that there are other kinds of "limits" that model theory can consider. For instance, the right kind of statement is true for $\mathbb{C}$ iff it is true for $\overline{\mathbb{F}}_p$ for arbitrarily large $p$. As I understand it, this was used to give the first proof that any injective polynomial map $\mathbb{C}^n \to \mathbb{C}^n$ is surjective. | |
Nov 4, 2010 at 7:50 | comment | added | BCnrd | Dear arsmath: Here is a specific example of what I have in mind. Let $R$ be the direct limit of a direct system of rings $R_i$, and let $X_i$ be a corresponding system of finitely presented schemes (compatible with base change along the directed system). Let $X$ be the corresponding $R$-scheme. Prove that if $X$ is $R$-flat (resp. $R$-proper) then so is $X_i \rightarrow {\rm{Spec}}(R_i)$ for large $i$. Can model-theoretic methods provide proofs? | |
Nov 4, 2010 at 7:45 | comment | added | BCnrd | Dear arsmath: There's nothing wrong with using model theory. My points are: (i) in practice one needs results like what Charles describes but with way more general ring extensions than those of algebraically closed fields, for which it seems unlikely that model theory is applicable, and (ii) he says the intervention of "large" extension fields is essential in some model-theoretic approaches, and it is an instructive exercise in basic scheme methods to see that one can make simple proofs of all such things without "large" field extns. So my emphasis is on alternative useful techniques. | |
Nov 4, 2010 at 7:22 | comment | added | arsmath | What's wrong with using model theory? You make it sound like it's cheating. | |
Nov 4, 2010 at 5:40 | comment | added | BCnrd | Dear Charles: Entirely precise versions of this principle are given in extraordinary generality in EGA IV$_3$, sections 8, 9, 11, 12, 17 without ever needing anything remotely model-theoretic, and the added generality is extremely useful in practice. If one only cares about the framework you describe, it can all be done in an elegant manner using nothing deeper than the Nullstellensatz and generic flatness results; this is a very instructive (and perhaps challenging) exercise to figure out. | |
Nov 4, 2010 at 0:28 | history | answered | Charles Staats | CC BY-SA 2.5 |