From e983e5a67c11f35e6565661d3ebf0b579c38d094 Mon Sep 17 00:00:00 2001 From: zeramorphic Date: Wed, 25 Oct 2023 12:54:43 +0100 Subject: [PATCH] Lectures 09B --- iii/alggeom/03_schemes.tex | 89 +++++++++++-- iii/cat/03_adjunctions.tex | 14 +++ iii/cat/04_limits.tex | 158 ++++++++++++++++++++++++ iii/cat/main.tex | 2 + iii/mtncl/02_quantifier_elimination.tex | 60 +++++++++ iii/mtncl/03_ultraproducts.tex | 0 iii/mtncl/main.tex | 2 + 7 files changed, 316 insertions(+), 9 deletions(-) create mode 100644 iii/cat/04_limits.tex create mode 100644 iii/mtncl/03_ultraproducts.tex diff --git a/iii/alggeom/03_schemes.tex b/iii/alggeom/03_schemes.tex index 66e42d72..9f62ef00 100644 --- a/iii/alggeom/03_schemes.tex +++ b/iii/alggeom/03_schemes.tex @@ -106,14 +106,85 @@ \subsection{Definitions and examples} \begin{definition} A \emph{scheme} is a ringed space \( (X, \mathcal O_X) \) where every point \( p \in X \) has a neighbourhood \( U_p \) such that the ringed space \( (U_p, \mathcal O_{U_p}) \) is isomorphic to some affine scheme. \end{definition} +% \item Let \( X = \Spec \mathbb C[x, y] \) and \( U = \qty{(x, y)}^c \). +% Then the scheme \( U \) is not an affine scheme. +% % exercise: why? +\begin{proposition} + Let \( X \) be a scheme, \( U \subseteq X \) an open set, and \( i : U \hookrightarrow X \) be the inclusion map. + Then, the ringed space \( (U, \mathcal O_U) \) is a scheme, where + \[ \mathcal O_U = \eval{\mathcal O_X}_U = i^{-1} \mathcal O_X \] +\end{proposition} +For example, take \( X = \Spec A \) and \( U = U_f \) for some \( f \in A \). +Then \( (U, \mathcal O_U) \cong (\Spec A_f, \mathcal O_{\Spec A_f}) \). +\begin{proof} + Let \( p \in U \subseteq X \). + Since \( X \) is a scheme, we can find \( \qty(V_p, \eval{O_X}_{V_p}) \) inside \( X \) with \( p \in V_p \), such that \( V_p \) is isomorphic to an affine scheme. + Then take \( V_p \cap U \subseteq U \) with structure sheaf given by the inclusion map. + Note that \( V_p \cap U \) may not be affine, but \( V_p \cong \Spec B \), and the distinguished opens in \( \Spec B \) form a basis. + This reduces the problem to the example of a distinguished open set above. +\end{proof} +\begin{definition} + \emph{Affine space} of dimension \( n \) over \( k \) is defined to be + \[ \mathbb A_k^n = \Spec k[x_1, \dots, x_n] \] +\end{definition} \begin{example} - \begin{enumerate} - \item The spectrum \( \Spec A \) of each ring \( A \) is a scheme. - \item Let \( X \) be a scheme and \( U \subseteq X \) an open set. - Take \( \mathcal O_U = \eval{\mathcal O_X}_U \) to be the structure sheaf of \( U \). - Thus \( U \) is a scheme: as the \( U_f \) form a basis, there is always a sufficiently small neighbourhood inside \( U \) isomorphic to an affine scheme. - \item Let \( X = \Spec \mathbb C[x, y] \) and \( U = \qty{(x, y)}^c \). - Then the scheme \( U \) is not an affine scheme. - % exercise: why? - \end{enumerate} + Let + \[ U = \mathbb A_k^{n^2} \setminus \qty{\det (x_{ij}) = 0} \] + which is the open set representing \( GL_n(k) \). + We will show that the multiplication map \( U \times U \to U \) is a morphism of schemes. +\end{example} +\begin{example} + Let \( U = \mathbb A_k^2 \setminus (x, y) \). + This is a scheme representing a plane without an origin. + We claim that \( U \) is not an affine scheme. + Suppose that \( U \) were affine; we aim to calculate \( \mathcal O_U(U) \). + Write + \[ U_x = \mathbb V(x)^c \subseteq \mathbb A_k^2;\quad U_y = \mathbb V(y)^c \subseteq \mathbb A_k^2 \] + These two open sets cover \( U \), and + \[ U_x \cap U_y = U_{xy} = \mathbb A_k^2 \setminus \mathbb V(xy) \] + Then, + \[ \mathcal O_U(U_x) = k[x,x^{-1},y];\quad \mathcal O_U(U_y) = k[x,y,y^{-1}];\quad \mathcal O_U(U_x \cap U_y) = k[x,x^{-1},y,y^{-1}] \] + The restriction maps \( \mathcal O_U(U_x) \to \mathcal O_U(U_{xy}) \) and \( \mathcal O_U(U_y) \to \mathcal O_U(U_{xy}) \) are the obvious ones. + By the sheaf axioms, + \[ \mathcal O_U(U) = k[x,x^{-1},y] \cap k[x,y,y^{-1}] \subseteq k[x,x^{-1},y,y^{-1}] \] + Thus, \( \mathcal O_U(U) = k[x,y] \). + This is a contradiction: one way to see this is that there exists a maximal ideal \( (x, y) \) in the ring of global sections in \( (U, \mathcal O_U) \) with empty vanishing locus. +\end{example} + +\subsection{Gluing sheaves} +% TODO: move? +Let \( X \) be a topological space with a cover \( \qty{U_\alpha} \). +Let \( \qty{\mathcal F_\alpha} \) be sheaves on \( \qty{U_\alpha} \), with isomorphisms +\[ \varphi_{\alpha\beta} : \eval{\mathcal F_\alpha}_{U_\alpha \cap U_\beta} \to \eval{\mathcal F_\beta}_{U_\alpha \cap U_\beta} \] +such that +\[ \varphi_{\alpha\alpha} = \id;\quad \varphi_{\alpha\beta} = \varphi_{\beta\alpha}^{-1};\quad \varphi_{\beta\gamma} \circ \varphi_{\alpha\beta} = \varphi_{\alpha\gamma} \] +The last equation is called the \emph{cocycle condition}. +This combination of conditions resembles the definition of an equivalence relation, with reflexivity, symmetry, and transitivity. + +We will construct a sheaf \( \mathcal F \) on \( X \). +Given \( V \subseteq X \) open, we define +\[ \mathcal F(V) = \qty{(s_\alpha) \in \prod_\alpha \mathcal F_\alpha(U_\alpha \cap V) \midd \varphi_{\alpha\beta}\qty(\eval{s_\alpha}_{V \cap U_\alpha \cap U_\beta}) = \eval{s_\beta}_{V \cap U_\alpha \cap U_\beta}} \] +\( \mathcal F \) is a presheaf. +Indeed, given \( (s_\alpha) \in \mathcal F(V) \) and \( W \subseteq V \) open, we take +\[ \eval{(s_\alpha)}_W = \qty(\res_{W \cap U_\alpha}^{V \cap U_\alpha}(s_\alpha))_\alpha \] +This lies in \( \mathcal F(W) \) by the sheaf axioms. +One check easily check that this is a sheaf. +\begin{proposition} + \( \eval{\mathcal F}_{U_\gamma} \) and \( \mathcal F_\gamma \) are canonically isomorphic as sheaves on \( U_\gamma \). +\end{proposition} +\begin{proof} + First, we construct a map \( \mathcal F_\gamma \to \eval{\mathcal F}_{U_\gamma} \). + Let \( V \subseteq U_\gamma \) and \( s \in \mathcal F_\gamma(V) \). + Define its image in \( \eval{\mathcal F}_{U_\gamma} \) to be + \[ \varphi_{\gamma\alpha}\qty(\eval{s}_{V \cap U_\alpha})_\alpha \] + We must check that this tuple lies in \( \eval{\mathcal F}_{U_\gamma}(V) = \mathcal F(V) \). + \[ \varphi_{\alpha\beta} \circ \varphi_{\gamma\alpha}\qty(\eval{s}_{V \cap U_\alpha \cap U_\beta}) = \varphi_{\gamma\beta}\qty(\eval{s}_{V \cap U_\alpha \cap U_\beta}) \] +\end{proof} + +\subsection{???} +\begin{example} + Let \( (X, \mathcal O_X) \) and \( (Y, \mathcal O_Y) \) be schemes with open sets \( U \subseteq X, V \subseteq Y \), and let \( \varphi : (U, \mathcal \eval{O_X}_U) \to (V, \mathcal \eval{O_Y}_V) \) be an isomorphism. + The topological spaces \( X, Y \) can be glued on \( U, V \) using \( \varphi \). + We can similarly glue the relevant sheaves together, thus gluing \( X \) and \( Y \) together as schemes. + Note that in this case, there is no cocycle condition. \end{example} diff --git a/iii/cat/03_adjunctions.tex b/iii/cat/03_adjunctions.tex index 91d77e8e..836c5c1a 100644 --- a/iii/cat/03_adjunctions.tex +++ b/iii/cat/03_adjunctions.tex @@ -335,5 +335,19 @@ \subsection{Reflections} For an abelian group \( A \), its set of torsion elements forms a subgroup \( A_t \), which is a torsion group. Any homomorphism from a torsion group to \( A \) must factor through \( A_t \). Thus \( A_t \) is the coreflection of \( A \) in the category of torsion abelian groups, and \( \faktor{A}{A_t} \) is the reflection of \( A \) in the category of torsion-free abelian groups. + \item The full subcategory \( \mathbf{KHaus} \) of compact Hausdorff spaces is reflective in the category \( \mathbf{Top} \) of topological spaces. + The left adjoint to the inclusion map is the \emph{Stone--\v{C}ech compactification} functor \( \beta \). + We will construct this functor using the special adjoint functor theorem, which is explored in the next section. + \item Recall that a subset \( C \) of a topological space \( X \) is called \emph{sequentially closed} if for every sequence \( x_n \in C \) converging to a limit \( x \in X \), we have \( x \in C \). + We say that \( X \) is a \emph{sequential space} if all sequentially closed subsets are closed. + The full subcategory \( \mathbf{Seq} \) of sequential spaces is coreflective in \( \mathbf{Top} \). + Given a space \( X \), let \( X_s \) denote the same set, but where the topology is such that all sequentially closed sets are also taken to be closed. + The identity map \( X_s \to X \) is continuous, and forms the counit of the adjunction. + \item The category \( \mathbf{Preord} \) of preorders is reflective in \( \mathbf{Cat} \). + The left adjoint maps a category \( \mathcal C \) to the quotient category \( \faktor{\mathbb C}{\sim} \) where \( \sim \) identifies all parallel pairs of morphisms. + \item Let \( X \) be a topological space. + Then the poset \( \Omega X \) of open sets in \( X \) is coreflective in the poset \( PX \), since if \( U \) is open and \( A \) is an arbitrary subset of \( X \), then \( U \subseteq A \) if and only if \( U \subseteq A^\circ \). + Thus the interior operator \( (-)^\circ \) is right adjoint to the inclusion \( \Omega X \to PX \). + Dually, the poset of closed sets is reflective in \( PX \); the closure operator \( \overline{(-)} \) is left adjoint to the inclusion. \end{enumerate} \end{example} diff --git a/iii/cat/04_limits.tex b/iii/cat/04_limits.tex new file mode 100644 index 00000000..dc68de62 --- /dev/null +++ b/iii/cat/04_limits.tex @@ -0,0 +1,158 @@ +\subsection{Cones over diagrams} +To formally define limits and colimits, we first need to define more precisely what is meant by a diagram in a category. +\begin{definition} + Let \( J \) be a category, which will almost always be small, and often finite. + A \emph{diagram} of shape \( J \) in a category \( \mathcal C \) is a functor \( D : J \to \mathcal C \). +\end{definition} +We call the objects \( D(j) \) the \emph{vertices} of the diagram, and the morphisms \( D(\alpha) \) the \emph{edges} of the diagram. +\begin{example} + Let \( J \) be the finite category + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGJ1bGxldCJdLFsxLDAsIlxcYnVsbGV0Il0sWzEsMSwiXFxidWxsZXQiXSxbMCwxLCJcXGJ1bGxldCJdLFswLDFdLFsxLDJdLFswLDJdLFswLDNdLFszLDJdXQ== +\[\begin{tikzcd} + \bullet & \bullet \\ + \bullet & \bullet + \arrow[from=1-1, to=1-2] + \arrow[from=1-2, to=2-2] + \arrow[from=1-1, to=2-2] + \arrow[from=1-1, to=2-1] + \arrow[from=2-1, to=2-2] +\end{tikzcd}\] + A diagram of shape \( J \) in \( \mathcal C \) is exactly a commutative square in \( \mathcal C \). + The diagonal arrow is required to make \( J \) into a category. +\end{example} +\begin{example} + Let \( J \) be the finite category + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGJ1bGxldCJdLFsxLDAsIlxcYnVsbGV0Il0sWzEsMSwiXFxidWxsZXQiXSxbMCwxLCJcXGJ1bGxldCJdLFswLDFdLFsxLDJdLFswLDNdLFszLDJdLFswLDIsIiIsMSx7Im9mZnNldCI6LTJ9XSxbMCwyLCIiLDEseyJvZmZzZXQiOjJ9XV0= +\[\begin{tikzcd} + \bullet & \bullet \\ + \bullet & \bullet + \arrow[from=1-1, to=1-2] + \arrow[from=1-2, to=2-2] + \arrow[from=1-1, to=2-1] + \arrow[from=2-1, to=2-2] + \arrow[shift left=1, from=1-1, to=2-2] + \arrow[shift right=1, from=1-1, to=2-2] +\end{tikzcd}\] + Then a diagram of shape \( J \) in \( \mathcal C \) is a square of objects in \( \mathcal C \) whose morphisms may or may not commute. +\end{example} +\begin{definition} + Let \( D \) be a diagram of shape \( J \) in \( \mathcal C \). + A \emph{cone over \( D \)} consists of an object \( C \in \ob \mathcal C \) called the \emph{apex} of the cone, together with morphisms \( \lambda_j : A \to D(j) \) called the \emph{legs} of the cone, such that all triangles of the following form commute. + % https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzAsMSwiRChqKSJdLFsyLDEsIkQoaicpIl0sWzAsMSwiXFxsYW1iZGFfaiIsMl0sWzEsMiwiRChcXGFscGhhKSIsMl0sWzAsMiwiXFxsYW1iZGFfe2onfSJdXQ== +\[\begin{tikzcd} + & A \\ + {D(j)} && {D(j')} + \arrow["{\lambda_j}"', from=1-2, to=2-1] + \arrow["{D(\alpha)}"', from=2-1, to=2-3] + \arrow["{\lambda_{j'}}", from=1-2, to=2-3] +\end{tikzcd}\] +\end{definition} +We can define the notion of a morphism between cones. +\begin{definition} + Let \( (A, \lambda_j), (B, \mu_j) \) be cones over a diagram \( D \) of shape \( J \) in \( \mathcal C \). + Then a \emph{morphism of cones} is a morphism \( f : A \to B \) such that all triangles of the following form commute. + % https://q.uiver.app/#q=WzAsMyxbMCwwLCJBIl0sWzIsMCwiQiJdLFsxLDEsIkQoaikiXSxbMCwxLCJmIl0sWzEsMiwiXFxtdV9qIl0sWzAsMiwiXFxsYW1iZGFfaiIsMl1d +\[\begin{tikzcd} + A && B \\ + & {D(j)} + \arrow["f", from=1-1, to=1-3] + \arrow["{\mu_j}", from=1-3, to=2-2] + \arrow["{\lambda_j}"', from=1-1, to=2-2] +\end{tikzcd}\] +\end{definition} +This makes the class of cones over a diagram \( D \) into a category, which will be denoted \( \operatorname{Cone}(D) \). +\begin{remark} + A cone over a diagram \( D \) with apex \( A \) is the same as a natural transformation from the constant diagram \( \Delta A \) to \( D \), as we can expand the commutative triangles into the following form. + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJBIl0sWzEsMCwiQSJdLFsxLDEsIkQoaicpIl0sWzAsMSwiRChqKSJdLFswLDEsIjFfQSJdLFsxLDIsIlxcbGFtYmRhX3tqJ30iXSxbMCwzLCJcXGxhbWJkYV9qIiwyXSxbMywyLCJEKFxcYWxwaGEpIiwyXV0= +\[\begin{tikzcd} + A & A \\ + {D(j)} & {D(j')} + \arrow["{1_A}", from=1-1, to=1-2] + \arrow["{\lambda_{j'}}", from=1-2, to=2-2] + \arrow["{\lambda_j}"', from=1-1, to=2-1] + \arrow["{D(\alpha)}"', from=2-1, to=2-2] +\end{tikzcd}\] + Note that \( \Delta \) is a functor \( \mathcal C \to [J, \mathcal C] \), and thus \( \operatorname{Cone}(D) \) is exactly the comma category \( (\Delta \downarrow D) \). +\end{remark} + +\subsection{Limits} +\begin{definition} + A \emph{limit} for a diagram \( D \) of shape \( J \) in \( \mathcal C \) is a terminal object in the category of cones over \( D \). + Dually, a \emph{colimit} for \( D \) is an initial object in the category of cones under \( D \). +\end{definition} +A cone under a diagram is often called a \emph{cocone}. +\begin{remark} + Using the fact that \( \operatorname{Cone}(D) = (\Delta \downarrow D) \) where \( \Delta : \mathcal C \to [J, \mathcal C] \), the category \( \mathcal C \) has limits for all diagrams of shape \( J \) if and only if \( \Delta \) has a right adjoint. +\end{remark} +\begin{example} + \begin{enumerate} + \item If \( J \) is the empty category, there is a unique diagram \( D \) of shape \( J \) in any category \( \mathcal C \). + Thus, a cone over this diagram is just an object in \( \mathcal C \), and morphisms of cones are just morphisms in \( \mathcal C \). + In particular, \( \operatorname{Cone}(D) \cong \mathcal C \), so a limit for \( D \) is a terminal object in \( \mathcal C \). + Dually, a colimit of the empty diagram is an initial object. + \item Let \( J \) be the discrete category with two objects. + A diagram of shape \( J \) in \( \mathcal C \) is thus a pair of objects. + A cone over this diagram is a \emph{span}. + \[\begin{tikzcd} + & C \\ + A && B + \arrow[from=1-2, to=2-1] + \arrow[from=1-2, to=2-3] + \end{tikzcd}\] + A limit cone is precisely a categorical product \( A \times B \). + \[\begin{tikzcd} + & {A \times B} \\ + A && B + \arrow["{\pi_1}"', from=1-2, to=2-1] + \arrow["{\pi_2}", from=1-2, to=2-3] + \end{tikzcd}\] + Similarly, the colimit for a pair of objects is a categorical coproduct \( A + B \). + \item If \( J \) is any discrete category, a diagram of shape \( J \) is a family of objects \( A_j \) in \( \mathcal C \) indexed by the objects of \( J \). + Limits and colimits over this diagram are products and coproducts of the \( A_j \). + \item If \( J \) is the category \( \bullet \rightrightarrows \bullet \), a diagram of shape \( J \) is a parallel pair of morphisms \( f, g : A \rightrightarrows B \). + A cone over such a parallel pair is + % https://q.uiver.app/#q=WzAsMyxbMSwwLCJDIl0sWzAsMSwiQSJdLFsyLDEsIkIiXSxbMCwxLCJoIiwyXSxbMCwyLCJrIl0sWzEsMiwiZiIsMCx7Im9mZnNldCI6LTJ9XSxbMSwyLCJnIiwyLHsib2Zmc2V0IjoyfV1d +\[\begin{tikzcd} + & C \\ + A && B + \arrow["h"', from=1-2, to=2-1] + \arrow["k", from=1-2, to=2-3] + \arrow["f", from=2-1, to=2-3] + \arrow["g"', shift right=2, from=2-1, to=2-3] +\end{tikzcd}\] + satisfying \( fh = k = gh \). + Equivalently, it is a morphism \( h : C \to A \) satisfying \( fh = gh \). + Thus, a limit is an equaliser, and dually, a colimit is a coequaliser. + \item Let \( J \) be the category + % https://q.uiver.app/#q=WzAsMyxbMSwwLCJcXGJ1bGxldCJdLFsxLDEsIlxcYnVsbGV0Il0sWzAsMSwiXFxidWxsZXQiXSxbMCwxXSxbMiwxXV0= +\[\begin{tikzcd} + & \bullet \\ + \bullet & \bullet + \arrow[from=1-2, to=2-2] + \arrow[from=2-1, to=2-2] +\end{tikzcd}\] + A diagram of shape \( J \) is thus a cospan in \( \mathcal C \). + % https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzEsMSwiQyJdLFswLDEsIkIiXSxbMCwxLCJmIl0sWzIsMSwiZyIsMl1d +\[\begin{tikzcd} + & A \\ + B & C + \arrow["f", from=1-2, to=2-2] + \arrow["g"', from=2-1, to=2-2] +\end{tikzcd}\] + A cone over this diagram is + % https://q.uiver.app/#q=WzAsNCxbMSwwLCJBIl0sWzEsMSwiQyJdLFswLDEsIkIiXSxbMCwwLCJEIl0sWzAsMSwiZiJdLFsyLDEsImciLDJdLFszLDAsImgiXSxbMywxLCJcXGVsbCIsMl0sWzMsMiwiayIsMl1d +\[\begin{tikzcd} + D & A \\ + B & C + \arrow["f", from=1-2, to=2-2] + \arrow["g"', from=2-1, to=2-2] + \arrow["h", from=1-1, to=1-2] + \arrow["\ell"', from=1-1, to=2-2] + \arrow["k"', from=1-1, to=2-1] +\end{tikzcd}\] + where \( \ell = fh = gk \) is redundant. + Thus a cone is a span that completes the commutative square. + A limit for the cospan is the universal way to complete this commutative square, which is called a \emph{pullback} of \( f \) and \( g \). + Dually, colimits of spans are called \emph{pushouts}. + \end{enumerate} +\end{example} diff --git a/iii/cat/main.tex b/iii/cat/main.tex index 0c5a7056..4110650b 100644 --- a/iii/cat/main.tex +++ b/iii/cat/main.tex @@ -16,5 +16,7 @@ \section{The Yoneda lemma} \input{02_yoneda_lemma.tex} \section{Adjunctions} \input{03_adjunctions.tex} +\section{Limits} +\input{04_limits.tex} \end{document} diff --git a/iii/mtncl/02_quantifier_elimination.tex b/iii/mtncl/02_quantifier_elimination.tex index bfe81c66..d08a9e48 100644 --- a/iii/mtncl/02_quantifier_elimination.tex +++ b/iii/mtncl/02_quantifier_elimination.tex @@ -473,3 +473,63 @@ \subsection{Interaction with other properties} \end{enumerate} Then \( \mathcal T \) has quantifier elimination. \end{corollary} +\begin{proof} + We show that condition (ii) of the theorem above holds. + Let \( \mathcal A, \mathcal B \) be models of \( \mathcal T \), and \( \vb a \in \mathcal A, \vb b \in \mathcal B \) be such that \( (\mathcal A, \vb a) \equiv_0 (\mathcal B, \vb b) \). + It suffices to show that \( (\mathcal A, \vb a) \Rightarrow_1 (\mathcal B, \vb b) \). + Let \( \varphi(\vb x, y) \) be quantifier-free, and such that \( \mathcal A \vDash \exists \vb y.\, \varphi(\vb a, \vb y) \). + Let \( \vb c = (c_0, \dots, c_{k-1}) \in \mathcal A \) be such a witness, so \( \mathcal A \vDash \varphi(\vb a, \vb c) \). + + We claim that there is an elementary extension \( \mathcal B_0 \) of \( \mathcal B \) and an element \( d_0 \in \mathcal B_0 \) such that \( (\mathcal A, \vb a, c_0) \equiv_0 (\mathcal B_0, \vb b, d_0) \). + If we can do this, we can iterate the process to obtain a chain of elementary extensions + \[ \mathcal B \preceq \mathcal B_0 \preceq \mathcal B_1 \preceq \dots \preceq \mathcal B_{k-1} \] + and elements \( d_i \in \mathcal B_i \) such that \( (\mathcal A, \vb a, \vb c) \equiv_0 (\mathcal B, \vb b, \vb d) \). + Then \( \mathcal B_{k-1} \vDash \varphi(\vb b, \vb d) \) as \( \varphi \) is quantifier-free, so \( \mathcal B_{k-1} \vDash \exists y.\, \varphi(\vb b, \vb y) \), giving \( \mathcal B \vDash \exists y.\, \varphi(\vb b, \vb y) \) as \( \mathcal B_{k-1} \equiv \mathcal B \) as required. + + To find \( \mathcal B_0 \) and \( d_0 \), we use the hypotheses and the compactness theorem. + As \( (\mathcal A, \vb a) \equiv_0 (\mathcal B, \vb b) \), there is an isomorphism \( \langle \vb a \rangle_{\mathcal A} \to \langle \vb b \rangle_{\mathcal B} \). + Take \( \mathcal C = \langle \vb a \rangle_{\mathcal A} \subseteq \mathcal A \). + By hypothesis (ii), there is an initial intermediate model \( \mathcal C \subseteq \mathcal A' \subseteq \mathcal A \) with \( \mathcal A' \vDash \mathcal T \), and there is an embedding \( \mathcal A' \hookrightarrow \mathcal B \) fixing \( \mathcal C \). + Without loss of generality, let us assume that this embedding is an inclusion. + Write + \[ \Psi = \qty{\psi(\vb x, y) \mid \mathcal A \vDash \psi(\vb a, c_0),\, \psi \text{ quantifier-free}} \] + As \( \vb a \in \mathcal A' \), we have that \( \mathcal A' \vDash \exists y.\, \psi(\vb a, y) \) for all \( \psi \in \Psi \) by hypothesis (a). + Now, \( \mathcal A' \subseteq \mathcal B \), and existential formulae are preserved under extension, so \( \mathcal B \vDash \exists y.\, \psi(\vb b, y) \) for all \( \psi \in \Psi \). + We conclude that every finite subset of \( \Psi \) is satisfied by some element of \( \mathcal B \), as finite conjunctions of quantifier-free formulae are also quantifier-free. + Thus, by compactness, there is an elementary extension \( \mathcal B \preceq \mathcal B_0 \) and \( d_0 \in \mathcal B_0 \) satisfying the formulae in \( \Psi \). + In particular, \( (\mathcal A, \vb a, c_0) \equiv_0 (\mathcal B_0, \vb b, d_0) \). +\end{proof} +\begin{example} + The theory \( \mathsf{RCF} \) of \emph{real closed fields} is the theory of ordered fields for which every nonnegative element is a square, and that all odd polynomials have a root. + Equivalently, it is the theory of ordered fields elementarily equivalent to \( \mathbb R \). + We show that this theory, with signature \( (+, \times, 0, 1, <) \), has quantifier elimination. + We will assume that every ordered field has a \emph{real closure}, and that a real closed field satisfies the intermediate value theorem for polynomials. + + We show that hypothesis (i) of the corollary above holds. + Suppose we have an embedding \( \mathcal A \subseteq \mathcal B \) of real closed fields, \( \vb a \in A \), and a quantifier-free formula \( \varphi(\vb x, y) \) such that \( \mathcal B \vDash \exists y.\, \varphi(\vb a, y) \). + By considering the disjunctive normal form, we may assume that \( \varphi \) is a disjunction of a conjunction of literals. + Moreover, the formulae \( y \neq z \) and \( y \nless z \) can be written in terms of \( = \) and \( < \). + Thus, we may assume that \( \varphi(\vb a, y) \) is of the form + \[ \qty(\bigwedge_{i < r} p_i(y) = 0) \vee \qty(\bigwedge_{j < s} 0 < q_j(y)) \] + where \( p_i, q_j \) are polynomials with coefficients in \( \mathcal A \). + If \( \varphi \) contains a nontrivial equation \( p_i(y) = 0 \), then if a witness exists in \( \mathcal B \), it must be algebraic over \( \mathcal A \). + One can show algebraically that this witness must lie in \( \mathcal A \). + Therefore, let us suppose \( r = 0 \). + + There are only finitely many points \( c_0, \dots, c_{n-1} \in \mathcal A \) that are roots for the \( q_j(y) \). + Since the real closed fields satisfy the intermediate value theorem for polynomials, the \( q_j(y) \) can only change sign at the \( c_i \). + Note that + \[ \mathcal A \vDash \forall x y.\, x < y \Rightarrow \exists z.\, (x < z \wedge z < y) \] + Since the \( c_i \) lie in \( \mathcal A \), there is an element of \( \mathcal A \) between any pair of distinct \( c_i \). + Suppose \( b \) witnesses \( \exists y.\, \varphi(\vb a, y) \) in \( \mathcal B \). + If there is a smallest interval \( (c_i, c_j) \) containing \( \mathcal B \), we can pick \( a \in \mathcal A \) also inside this interval, giving \( \mathcal A \vDash \varphi(\vb a, a) \) as required. + The other cases are similar. + + We now show hypothesis (ii). + Suppose \( \mathcal C \subseteq \mathcal A \) where \( \mathcal A \) is a real closed field. + Then \( \mathcal C \) is an ordered integral domain. + The field of fractions of \( \mathcal C \) can be made an ordered field in a canonical way, by saying \( \frac{a}{b} > 0 \) if \( ab > 0 \). + The embedding \( \mathcal C \) into \( \mathcal A \) is an injective homomorphism of ordered rings, into an ordered field. + By the universal property of the fraction field, there is a unique homomorphism of ordered fields from \( FF(\mathcal C) \) to \( \mathcal A \) that extends the inclusion of \( \mathcal C \) into \( \mathcal A \). + Let \( \mathcal A' \) be the real closure of \( FF(\mathcal C) \). +\end{example} diff --git a/iii/mtncl/03_ultraproducts.tex b/iii/mtncl/03_ultraproducts.tex new file mode 100644 index 00000000..e69de29b diff --git a/iii/mtncl/main.tex b/iii/mtncl/main.tex index 9c168754..61299db3 100644 --- a/iii/mtncl/main.tex +++ b/iii/mtncl/main.tex @@ -14,6 +14,8 @@ \section{Substructures} \input{01_substructures.tex} \section{Quantifier elimination} \input{02_quantifier_elimination.tex} +\section{Ultraproducts} +\input{03_ultraproducts.tex} \end{document}