#scheme theory

[just-evo-now]

I am delighted by this lecture series

[eatsleepmath]

Quantifier elimination extends to co-slice categories

Let \(T\) be a first-order theory which admits quantifier-elimination, and let \(K \subseteq A \in \operatorname{Mod}(T)\) be a substructure of some model of \(T\). Then the theory \[T \cup \mathsf{Diag}(K)\] (which is precisely the theory of models of \(T\) that admit an embedding from \(K\)) admits quantifier elimination.

Proof. Let \(M\) and \(N\) be models of \(T\) that share a common substructure \(L\) which furthermore contains a copy of \(K\). Showing substructure-completeness (hence quantifier-elimination) means showing that \(M\) and \(N\) are \((K \to L)\)-equivalent in the language of \(T \cup \mathsf{Diag}(K \to L)\). Let \(\varphi(a)\) be a sentence in the language of \(T\) enriched by \(K\), hence a formula in the language of \(T\) with parameters \(a \in K \subseteq L\). We can then write \(\varphi(a) \equiv \widetilde{\varphi}(a,b)\) where \(b\) names the parameters from \(K\). As we can replace \(a,b\) with their corresponding names in the language of \(T \cup \mathsf{Diag}(L)\) to produce a sentence \(\psi(a,b)\) in that language, \[M \models \varphi(a) \iff M \models \psi(a,b) \overset{\text{q.e}}{\iff} N \models \psi(a,b) \iff N \models \varphi(a),\] which gives quantifier-elimination, as required. \(_\square\)

This (near-tautological result) seems to hint towards a scheme-theoretic connection: any categorical consequences of the fact that a class of structures admits QE extends also to co-slice categories, e.g. not only do algebraically closed fields admit QE, but so do algebraically closed fields under \(R\), which correspond to the geometric points of \(\operatorname{Spec}(R)\), c.f. a recent expository paper by Betts on quantifier elimination and Chevalley’s theorem in the abstract, scheme-theoretic setting.

[isomorphismes]

> [blowup](https://en.wikipedia.org/wiki/Blowing_up) of the projective plane ℙ²

by [Charles Staats](http://tex.stackexchange.com/users/484) HT [@hdevalence](https://twitter.com/hdevalence/status/432734193722744832) [](http://math.stackexchange.com/a/97448/1457)

[isomorphismes]

What are the definition and motivation of the mathematical concepts of schemes and sheaves?

Answer by Alain Debecker on Quora:

A Sheaf is an object on which you can do differential geometry and a Scheme is an object on which you can do algebraic geometry.   Sheaf is a generalization of a manifold, which is a topological space locally isomorphic to an euclidean space, that is each point has a neighborhood that is homeomorphic to a (fixed) Euclidean space. For a sheaf you replace the Euclidean space by a (fixed) algebraic structure (ring/group/vector space/algebra, none,…) and homeomorphisms by another functional property (smooth, analytic, measurable, none…) A Scheme is a generalization of an algebraic variety. Instead of looking at a curve as a set of points, you look at the algebra of rational functions defined on the curve. Then a point x of the curve can be identified with functions such that ƒ(x)=0, that is, with a prime ideal. A scheme is a topological space locally isomorphic with such a beast.