diff --git a/iii/alggeom/03_schemes.tex b/iii/alggeom/03_schemes.tex index 913f6a0..cf3d620 100644 --- a/iii/alggeom/03_schemes.tex +++ b/iii/alggeom/03_schemes.tex @@ -244,8 +244,6 @@ \subsection{Gluing schemes} % exercise: \mathcal O_{\mathbb P^n_A}(\mathbb P^n_A) = A. \subsection{The Proj construction} -% Idea: -% Spec : Rings -> Schemes; Proj : GradedRings -> Schemes \begin{definition} A \emph{\( \mathbb Z \)-grading} on a ring \( A \) is a decomposition \[ A = \bigoplus_{i \in \mathbb Z} A_i \] @@ -260,4 +258,65 @@ \subsection{The Proj construction} % how? \end{example} Note that by definition, \( A_0 \) is a subring of \( A \). -We will always assume that the degree 1 elements of a graded ring generate \( A \) as an algebra over \( A_0 \). +For simplicity, we will always assume in this course that the degree 1 elements of a graded ring generate \( A \) as an algebra over \( A_0 \). +We also typically assume that \( A_i = 0 \) for \( i < 0 \). +We define +\[ A_+ = \bigoplus_{i \geq 1} A_i \subseteq A \] +This forms an ideal in \( A \), called the \emph{irrelevant ideal}. +If \( A \) is a polynomial ring with the usual grading, the irrelevant ideal corresponds to the point \( \vb 0 \) in the theory of varieties. +This aligns with the definition of projective space in classical algebraic geometry, in which the point \( \vb 0 \) is deleted. + +A \emph{homogeneous element} \( f \in A \) is an element contained in some \( A_d \). +An ideal \( I \) of \( A \) is called \emph{homogeneous} if it is generated by homogeneous elements. +\begin{definition} + Let \( A \) be a graded ring. + \( \Proj A \) is the set of homogeneous prime ideals in \( A \) that do not contain the irrelevant ideal. + If \( I \subseteq A \) is homogeneous, we define + \[ \mathbb V(I) = \qty{\mathfrak p \in \Proj A \mid I \subseteq \mathfrak p} \] + The \emph{Zariski topology} on \( \Proj A \) is the topology where the closed sets are of the form \( \mathbb V(I) \) where \( I \) is a homogeneous ideal. +\end{definition} +The Spec construction allows us to convert rings into schemes; the Proj construction allows us to convert graded rings into schemes. +Unlike Spec, the construction of Proj is not functorial. + +Let \( f \in A_1 \) and \( U_f = \Proj A \setminus \mathbb V(f) \). +Observe that the set \( \qty{U_f}_{f \in A_1} \) covers \( \Proj A \), because the \( f \) generate the unit ideal. +The ring \( A\qty[\frac{1}{f}] = A_f \) is naturally \( \mathbb Z \)-graded by defining \( \deg \frac{1}{f} = -\deg f \). +Note that \( A_f \) may have negatively graded elements, even though \( A \) does not. +\begin{example} + Let \( A = k[x_0, x_1] \) and \( f = x_0 \). + Then in \( A\qty[\frac{1}{f}] = k[x_0, x_1, x_0^{-1}] \), the degree zero elements include \( k \) and elements such as \( \frac{x_1}{x_0}, \frac{x_1^2 + x_1 x_0}{x_0^2} \). + There are degree one elements such as \( \frac{x_1^2}{x_0} \). +\end{example} +\begin{proposition} + There is a natural bijection + \[ \qty{\text{homogeneous prime ideals in \( A \) that miss \( f \)}} \leftrightarrow \qty{\text{prime ideals in } (A_f)_0} \] +\end{proposition} +Note also that the set of homogeneous prime ideals in \( A \) that miss \( f \) are naturally in bijection with the homogeneous prime ideals in \( A_f \). +\begin{proof} + Suppose \( \mathfrak q \) is a prime ideal in \( \qty(A\qty[\frac{1}{f}])_0 \). + Then let \( \psi(\mathfrak q) \) be the ideal + \[ \psi(\mathfrak q) = \qty( \bigcup_{d \geq 0} \qty{a \in A_d \midd \frac{a}{f^d} \in \mathfrak q} \subseteq A ) \] + One can check that this is prime. + Now suppose \( \mathfrak p \) is a homogeneous prime ideal missing \( f \). + Define \( \varphi(\mathfrak p) \) to be + \[ \varphi(\mathfrak p) = \qty(p \cdot A\qty[\frac{1}{f}] \cap \qty(A\qty[\frac{1}{f}])_0) \] + This ideal is also prime. + + One can easily check that \( \varphi \circ \psi \) is the identity. + For the other direction, suppose \( \mathfrak p \) is a homogeneous prime ideal missing \( f \); we show that \( \mathfrak p = \psi(\varphi(\mathfrak p)) \) by antisymmetry. + If \( a \in \mathfrak p \in A_d \), then \( \frac{a}{f^d} \in \varphi(\mathfrak p) \), so \( a \in \psi(\varphi(\mathfrak p)) \) by construction. + Conversely, if \( a \in \psi(\varphi(\mathfrak p)) \), then \( \frac{a}{f^d} \in \varphi(\mathfrak p) \) for some \( d \), so there exists \( b \in \mathfrak p \) such that \( \frac{b}{f^e} = \frac{a}{f^d} \) in \( A\qty[\frac{1}{f}] \). + Hence for some \( k \geq 0 \), we have \( f^k (f^d b - f^e a) = 0 \), and \( f^{e+k} \notin \mathfrak p \). + But by primality, \( a \in \mathfrak p \), as required. +\end{proof} +The bijection constructed is compatible with ideal containment, so is a homeomorphism of topological spaces +\[ U_f \leftrightarrow \Spec (A_f)_0 \] +Thus \( \Proj A \) is covered by open sets homeomorphic to an affine scheme. +If \( f, g \in A_1 \), then \( U_f \cap U_g \) is naturally homeomorphic to +\[ \qty(\Spec A\qty[\frac{1}{f}])_0\qty[\frac{f}{g}] = \Spec \qty(A\qty[f^{-1}, g^{-1}])_0 \] +Take the open cover \( \qty{U_f} \) with structure sheaf \( \mathcal O_{\Spec (A_f)_0} \) on each \( U_f \), and isomorphisms on \( U_f \cap U_g \) by the condition above. +The cocycle condition follows from the formal properties of the localisation. +Therefore, \( \Proj A \) is a scheme. + +If \( A = k[x_0, \dots, x_n] \) with the standard grading, we write \( \mathbb P^n_k \) for \( \Proj A \). + diff --git a/iii/alggeom/04_morphisms.tex b/iii/alggeom/04_morphisms.tex new file mode 100644 index 0000000..0c6e5e9 --- /dev/null +++ b/iii/alggeom/04_morphisms.tex @@ -0,0 +1,8 @@ +\subsection{???} +Let \( (X, \mathcal O_X) \) be a scheme. +The stalks \( \mathcal O_{X,\mathfrak p} \) are local rings: they have a unique maximal ideal, which is the set of all non-unit elements. +Given \( f \in \mathcal O_X(U) \), we can meaningfully ask whether \( f \) vanishes at \( \mathfrak p \); that is, if the image of \( f \) in \( \mathcal O_{X, \mathfrak p} \) is contained in the maximal ideal. +\begin{definition} + A morphism of ringed spaces \( f : (X, \mathcal O_X) \to (Y, \mathcal O_Y) \) consists of a continuous function \( f : X \to Y \) and a morphism \( f^\sharp : \mathcal O_Y \to f_\star \mathcal O_X \) between sheaves of rings on \( Y \). +\end{definition} +\( f^\sharp \) represents function composition with \( f^{-1} \), although the ring \( \mathcal O_X \) may not be a ring of functions. diff --git a/iii/alggeom/main.tex b/iii/alggeom/main.tex index 6d347a8..23b81a3 100644 --- a/iii/alggeom/main.tex +++ b/iii/alggeom/main.tex @@ -18,5 +18,7 @@ \section{Sheaves} \input{02_sheaves.tex} \section{Schemes} \input{03_schemes.tex} +\section{Morphisms} +\input{04_morphisms.tex} \end{document} diff --git a/iii/cat/04_limits.tex b/iii/cat/04_limits.tex index 80c0c23..9ce5ea8 100644 --- a/iii/cat/04_limits.tex +++ b/iii/cat/04_limits.tex @@ -281,3 +281,108 @@ \subsection{Preservation and creation} A free product of two groups \( G, H \) is always nonabelian, except for the case where either \( G \) or \( H \) is the trivial group, but the coproduct of the trivial group with \( H \) is isomorphic to \( H \) in both categories. \end{enumerate} \end{example} +\begin{lemma} + Suppose \( \mathcal D \) has limits of shape \( J \). + Then, for any \( \mathcal C \), the functor category \( [\mathcal C, \mathcal D] \) also has limits of shape \( J \), and the forgetful functor \( [\mathcal C, \mathcal D] \to \mathcal D^{\ob \mathcal C} \) creates them. +\end{lemma} +\begin{proof} + Given a diagram \( D : J \to [\mathcal C, \mathcal D] \), we can regard it as a functor \( D : J \times \mathcal C \to \mathcal D \), so for a fixed object in \( \mathcal C \), we obtain a diagram \( D(-, A) \) of shape \( J \) in \( \mathcal D \), which has a limit \( (LA, (\lambda_{j,A})_{j \in \ob J}) \). + Given any \( f : A \to B \) in \( \mathcal C \), the composites + % https://q.uiver.app/#q=WzAsMyxbMCwwLCJMQSJdLFsxLDAsIkQoaixBKSJdLFsyLDAsIkQoaixCKSJdLFswLDEsIlxcbGFtYmRhX3tqLEF9Il0sWzEsMiwiRChqLGYpIl1d +\[\begin{tikzcd} + LA & {D(j,A)} & {D(j,B)} + \arrow["{\lambda_{j,A}}", from=1-1, to=1-2] + \arrow["{D(j,f)}", from=1-2, to=1-3] +\end{tikzcd}\] + form a cone over \( D(-, B) \), and so factor uniquely through its limit \( LB \). + Thus we obtain \( Lf : LA \to LB \). + This is functorial because \( Lf \) is unique with this property. + This is the unique lifting of \( (LA)_{A \in \ob \mathcal C} \) to an object of \( [\mathcal C, \mathcal D] \) which makes the \( \lambda_{j,-} \) into natural transformations. + It is a limit cone in \( [\mathcal C, \mathcal D] \): given any cone in \( [\mathcal C, \mathcal D] \) with apex \( M \) and legs \( (\mu_{j,-})_{j \in \ob J} \) over \( D \), the \( \mu_{j,A} \) form a cone over \( D(-, A) \), so we obtain a unique \( \nu_A : MA \to LA \) such that \( \lambda_{j,A} \nu_A = \mu_{j,A} \) for all \( A \). + The \( \nu_A \) form a natural transformation \( M \to L \), because for any \( f : A \to B \) in \( \mathcal C \), the two paths \( \nu_B(Mf), (Lf)\nu_A : MA \rightrightarrows LB \) are factorisations of the same cone over \( D(-, B) \) through its limit, so must be equal. +\end{proof} +\begin{remark} + Note that \( f : A \to B \) is monic if and only if +\[\begin{tikzcd} + A & A \\ + A & B + \arrow["{1_A}", from=1-1, to=1-2] + \arrow["f", from=1-2, to=2-2] + \arrow["{1_A}"', from=1-1, to=2-1] + \arrow["f"', from=2-1, to=2-2] +\end{tikzcd}\] + is a pullback square. + Thus, if \( \mathcal D \) has pullbacks, any monomorphism in \( [\mathcal C, \mathcal D] \) is a pointwise monomorphism, because the pullback in \( [\mathcal C, \mathcal D] \) is constructed pointwise by the previous lemma. +\end{remark} + +\subsection{Adjoint functor theorems} +\begin{lemma} + Let \( G : \mathcal D \to \mathcal C \) be a functor with a left adjoint. + Then \( G \) preserves all limits which exist in \( \mathcal D \). +\end{lemma} +\begin{proof}[Proof 1] + In this proof, we will assume that \( \mathcal C, \mathcal D \) both have all limits of shape \( J \). + If \( F \dashv G \), then the diagram + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXG1hdGhjYWwgQyJdLFsxLDAsIlxcbWF0aGNhbCBEIl0sWzAsMSwiW0osIFxcbWF0aGNhbCBDXSJdLFsxLDEsIltKLCBcXG1hdGhjYWwgRF0iXSxbMCwxLCJGIl0sWzAsMiwiXFxEZWx0YSIsMl0sWzEsMywiXFxEZWx0YSJdLFsyLDMsIltKLEZdIiwyXV0= +\[\begin{tikzcd} + {\mathcal C} & {\mathcal D} \\ + {[J, \mathcal C]} & {[J, \mathcal D]} + \arrow["F", from=1-1, to=1-2] + \arrow["\Delta"', from=1-1, to=2-1] + \arrow["\Delta", from=1-2, to=2-2] + \arrow["{[J,F]}"', from=2-1, to=2-2] +\end{tikzcd}\] + commutes. + All of the functors in this diagram have right adjoints, so the diagram + % https://q.uiver.app/#q=WzAsNCxbMSwxLCJbSiwgXFxtYXRoY2FsIERdIl0sWzEsMCwiXFxtYXRoY2FsIEQiXSxbMCwwLCJcXG1hdGhjYWwgQyJdLFswLDEsIltKLCBcXG1hdGhjYWwgQ10iXSxbMCwxLCJcXGxpbV9KIiwyXSxbMSwyLCJHIiwyXSxbMCwzLCJbSiwgR10iXSxbMywyLCJcXGxpbV9KIl1d +\[\begin{tikzcd} + {\mathcal C} & {\mathcal D} \\ + {[J, \mathcal C]} & {[J, \mathcal D]} + \arrow["{\lim_J}"', from=2-2, to=1-2] + \arrow["G"', from=1-2, to=1-1] + \arrow["{[J, G]}", from=2-2, to=2-1] + \arrow["{\lim_J}", from=2-1, to=1-1] +\end{tikzcd}\] + commutes up to natural isomorphism, where \( \lim_J \) sends a diagram of shape \( J \) to the apex of its limit cone. + But this is exactly the statement that \( G \) preserves limits. +\end{proof} +\begin{proof}[Proof 2] + In this proof, we will not assume that \( \mathcal C \) has limits of any kind, and only assume a single diagram \( D : J \to \mathcal D \) has a limit cone \( (L, (\lambda_j)_{j \in \ob J}) \) over it. + Given any cone over \( GD \) with apex \( A \) and legs \( \mu_j : A \to GD(j) \), the legs correspond under the adjunction to morphisms \( \overline \mu_j : FA \to D(j) \), which form a cone over \( D \) by naturality of the adjunction. + We obtain a unique factorisation \( \overline \mu : FA \to L \) with \( \lambda_j \overline \mu = \overline \mu_j \) for all \( j \), or equivalently, \( (G\lambda_j)\mu = \mu_j \), where \( \mu : A \to GL \) corresponds to \( \overline \mu \) under the adjunction. +\end{proof} +Suppose that \( \mathcal D \) has and \( G : \mathcal D \to \mathcal C \) preserves all limits. +We would expect \( G \) to have a left adjoint. +\begin{lemma} + Suppose that \( \mathcal D \) has and \( G : \mathcal D \to \mathcal C \) preserves limits of shape \( J \). + Then for any \( A \in \ob \mathcal C \), the category \( (A \downarrow G) \) has limits of shape \( J \), and the forgetful functor \( U : (A \downarrow G) \to \mathcal D \) creates them. +\end{lemma} +\begin{proof} + Let \( D : J \to (A \downarrow G) \) be a diagram. + We write each \( D(j) \) as \( (UD(j), f_j) \) where \( f_j : A \to GUD(j) \). + Let \( (L, (\lambda_j)_{j \in \ob J}) \) be a limit for \( UD \) in \( \mathcal D \). + By assumption, \( (GL, (G\lambda_j)_{j \in \ob J}) \) is a limit for \( GUD \) in \( \mathcal C \). + But the edges of \( \mathcal D \) are morphisms in \( (A \downarrow G) \), so the \( f_j \) form a cone over \( GUD \). + Thus, we obtain a unique factorisation \( f : A \to GL \) such that \( (G\lambda_j) f = f_j \) for all \( j \). + In other words, we have a unique lifting of \( L \) to an object \( (L, f) \) of \( (A \downarrow G) \) which makes the \( \lambda_j \) into a cone over \( D \) with apex \( (L, f) \). + Any cone over \( \mathcal D \) with apex \( (M, g) \), becomes a cone over \( UD \) with apex \( M \) by forgetting the structure map, so we get a unique \( h : M \to L \), and this becomes a morphism in \( (A \downarrow G) \) as both \( (Gh)g \) and \( f \) are factorisations through \( L \) of the same cone over \( UD \). +\end{proof} +\begin{lemma} + Let \( \mathcal C \) be a category. + Specifying an initial object of \( \mathcal C \) is equivalent to specifying a limit for the identity functor \( 1_{\mathcal C} : \mathcal C \to \mathcal C \), considered as a diagram of shape \( \mathcal C \) in \( \mathcal C \). +\end{lemma} +\begin{proof} + First, suppose we have an initial object \( I \) in \( \mathcal C \). + Then the unique morphisms \( I \to A \) form a cone over \( 1_{\mathcal C} \), and it is a limit, because for any other cone \( (B, (\lambda_A : B \to A)) \), then \( \lambda_I \) is the unique factorisation as required. + Conversely, suppose \( (I, (\lambda_A : I \to A)) \) is a limit for \( 1_{\mathcal C} \). + Then certainly \( I \) is \emph{weakly initial}: it has at least one morphism to any other object, given by \( \lambda_A \). + For any morphism \( f : I \to A \), it is an edge of the diagram, so \( f \lambda_I = \lambda_A \), so it suffices to show that \( \lambda_I \) is the identity morphism. + Using the same equation with \( f = \lambda_A \), we obtain \( \lambda_A \lambda_I = \lambda_A \), so \( \lambda_I \) is a factorisation of the limit cone through itself. + As this factorisation must be unique, we must have \( \lambda_I = 1_I \). +\end{proof} +\begin{proposition}[primitive adjoint functor theorem] + If \( \mathcal D \) has and \( G : \mathcal D \to \mathcal C \) preserves all limits, then \( G \) has a left adjoint. +\end{proposition} +\begin{proof} + The categories \( (A \downarrow G) \) have all limits, and in particular they have initial objects, so \( G \) has a left adjoint. +\end{proof} diff --git a/iii/mtncl/03_ultraproducts.tex b/iii/mtncl/03_ultraproducts.tex index 7cae99a..c972687 100644 --- a/iii/mtncl/03_ultraproducts.tex +++ b/iii/mtncl/03_ultraproducts.tex @@ -63,3 +63,90 @@ \subsection{Filters} \end{definition} For property (ii), we might also say that \( \mathcal F \) is a \emph{terminal segment} of \( X \). % large things get "stuck in the filter"; filters measure largeness +\begin{example} + \begin{enumerate} + \item Given an element \( j \in I \), the family \( \mathcal F_j \) of all subsets of \( I \) containing \( j \) is a filter on \( \mathcal P(I) \). + A filter of this form is called \emph{principal}. + A filter that is not principal is called \emph{free}. + \item The family of all cofinite subsets of \( I \) forms a filter on \( \mathcal P(I) \), called the \emph{Fr\'echet filter}. + One can show that any free maximal filter must contain the Fr\'echet filter. + % why? + \item The family of measurable subsets of \( [0,1] \) with Lebesgue measure \( 1 \) is a filter. + \end{enumerate} +\end{example} +\begin{definition} + A filter \( \mathcal F \) on a lattice \( L \) is \emph{proper} if it is not equal to \( L \). + A maximal proper filter is called an \emph{ultrafilter}. +\end{definition} +The ultrafilters on \( \mathcal P(I) \) are precisely those filters \( \mathcal F \) where for each \( U \subseteq I \), either \( U \in \mathcal F \) or \( I \setminus U \in \mathcal F \). +\begin{proposition}[the ultrafilter principle] + Given a set \( I \), every proper filter on \( \mathcal P(I) \) can be extended to an ultrafilter. +\end{proposition} +The ultrafilter principle is a choice principle that is strictly weaker than the axiom of choice. +\begin{proof} + Apply Zorn's lemma. + % TODO: flesh out? +\end{proof} + +\subsection{\L{}o\'s' theorem} +For \( \bm \alpha \in \prod_{i \in I} \mathcal M_i \) and \( \varphi(\vb x) \) an \( \mathcal L \)-formula, we write +\[ [\varphi(\bm \alpha)] = \qty{i \in I \mid \mathcal M_i \vDash \varphi(\bm \alpha(i))} \] +Let \( I \) be a set and \( \mathcal F \) be a filter on \( \mathcal P(I) \). +Let \( \qty{\mathcal M_i}_{i \in I} \) be a family of \( \mathcal L \)-structures. +The carrier set for the reduced product \( \faktor{\prod \mathcal M_i}{\mathcal F} \) is the quotient of the cartesian product \( \prod_{i \in I} \mathcal M_i \) by the equivalence relation defined by \( \alpha \sim \beta \) if and only if \( [\alpha = \beta] \in \mathcal F \). +We write \( \langle \alpha \rangle \) for the equivalence class of \( \alpha \) in the reduced product. +If \( \mathcal F \) is an ultrafilter, we call the reduced product an \emph{ultraproduct}. +If all of the factors \( \mathcal M_i \) are equal, the ultraproduct is called an \emph{ultrapower}. + +We turn the reduced product into an \( \mathcal L \)-structure as follows. +\[ f^{\faktor{\prod \mathcal M_i}{\mathcal F}}(\langle \alpha_1 \rangle, \dots, \langle \alpha_n \rangle) = \langle \lambda i.\, f^{\mathcal M_i}(\alpha_1(i), \dots, \alpha_n(i)) \rangle \] +\[ (\langle \alpha_1 \rangle, \dots, \langle \alpha_n \rangle) \in R^{\faktor{\prod \mathcal M_i}{\mathcal F}} \iff [R(\alpha_1, \dots, \alpha_n)] \in \mathcal F \] +Note that if \( \mathcal F = \mathcal F_j \) is a principal filter, then \( \faktor{\prod \mathcal M_i}{\mathcal F} \cong \mathcal M_j \). + +\begin{theorem} + Let \( \qty{\mathcal M_i}_{i \in I} \) be a set of \( \mathcal L \)-structures, and \( \mathcal U \) be an ultrafilter on \( \mathcal P(I) \). + Then for all \( (\langle \alpha_1 \rangle, \dots, \langle \alpha_n \rangle) \in \qty(\faktor{\prod \mathcal M_i}{\mathcal U})^n \) and \( \mathcal L \)-formulae \( \varphi(x_1, \dots, x_n) \), + \[ \faktor{\prod \mathcal M_i}{\mathcal U} \vDash \varphi(\langle \alpha_1 \rangle, \dots, \langle \alpha_n \rangle) \iff [\varphi(\alpha_1, \dots, \alpha_n)] \in \mathcal U \] +\end{theorem} +In particular, if each \( \mathcal M_i \) is a model for some theory \( \mathcal T \), then so is the ultraproduct. +\begin{proof} + We prove the result by induction on the length of \( \varphi \). + The result holds for atomic formulae by the definition of the interpretations of function and relation symbols. + Since all first-order formulae are equivalent to one composed of atomic formulae under negations, conjunctions, and existential quantification, it suffices to check these cases. + + If the theorem holds for \( \psi \), and \( \varphi = \neg \psi \), we can negate both sides of the induction hypothesis to show that + \[ \faktor{\prod \mathcal M_i}{\mathcal U} \vDash \neg\psi \iff [\psi] \notin \mathcal U \] + As \( \mathcal U \) is an ultrafilter, the right hand side holds if and only if the complement of \( [\psi] \) lies in \( \mathcal U \). + But this complement is precisely \( [\neg\psi] \), as required. + + If the theorem holds for \( \psi_1, \psi_2 \), then + \[ \faktor{\prod \mathcal M_i}{\mathcal U} \vDash \psi_i \iff [\psi_i] \in \mathcal U \] + \begin{align*} + \faktor{\prod \mathcal M_i}{\mathcal U} \vDash \psi_1 \wedge \psi_2 &\iff [\psi_1] \in \mathcal U \text{ and } [\psi_2] \in \mathcal U \\ + &\iff [\psi_1 \wedge \psi_2] \in \mathcal U + \end{align*} + Indeed, if \( [\psi_1 \wedge \psi_2] \in \mathcal U \), then both \( [\psi_1] \) and \( [\psi_2] \) are in \( \mathcal U \), since \( [\psi_1 \wedge \psi_2] \subseteq [\psi_1], [\psi_2] \). + Conversely, if \( [\psi_1], [\psi_2] \in \mathcal U \), then \( [\psi_1] \cap [\psi_2] \subseteq [\psi_1 \wedge \psi_2] \) as they are equal, but \( [\psi_1] \cap [\psi_2] \in \mathcal U \), so \( [\psi_1 \wedge \psi_2] \in \mathcal U \). + + For the case of existential quantification, we will use the axiom of choice. + Let \( x \) be free in \( \psi \). + We have + \[ \faktor{\prod \mathcal M_i}{\mathcal U} \vDash \exists x.\, \psi(x) \iff \exists \langle \alpha \rangle.\, \faktor{\prod \mathcal M_i}{\mathcal U} \vDash \psi(\langle \alpha \rangle) \] + By the inductive hypothesis, the right hand side holds if and only if \( [\psi(\alpha)] \in \mathcal U \). + Suppose that + \[ \faktor{\prod \mathcal M_i}{\mathcal U} \vDash \psi(\langle \alpha \rangle) \] + Then \( [\psi(\alpha)] \subseteq [\exists x.\, \psi(x)] \in \mathcal U \), as \( \mathcal U \) is a filter. + + Conversely, suppose \( [\exists x.\, \psi(x)] \in \mathcal U \). + Using the axiom of choice, we can choose a witness \( \alpha(i) \) to \( \mathcal M_i \vDash \exists x.\, \psi(x) \) for each \( i \in [\exists x.\, \psi(x)] \). + For each \( i \notin [\exists x.\, \psi(x)] \), we choose an arbitrary element of \( \mathcal M_i \). + Hence, + \[ \faktor{\prod \mathcal M_i}{\mathcal U} \vDash \psi(\langle \alpha \rangle) \] +\end{proof} +\begin{remark} + \begin{enumerate} + \item Since \( \mathcal U \) is an ultrafilter, the complement of \( [\exists x.\, \psi(x)] \) is not in \( \mathcal U \). + Thus, the set of indices \( I \) for which \( \alpha(i) \) was chosen arbitrarily does not lie in the ultrafilter, so this choice does not change the equivalence class of \( \alpha \). + \item The use of the axiom of choice in the above theorem is essential. + \end{enumerate} +\end{remark} diff --git a/util.sty b/util.sty index fd7fd07..df2cd69 100644 --- a/util.sty +++ b/util.sty @@ -215,6 +215,7 @@ \DeclareMathOperator{\mor}{mor} \DeclareMathOperator{\cod}{cod} \DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\Proj}{Proj} \DeclareMathOperator{\res}{res} % https://github.com/wspr/unicode-math/issues/457 \AtBeginDocument{%