Some edits

Signed-off-by: zeramorphic <[email protected]>

iii/commalg/02_tensor_products.tex

@@ -56,7 +56,7 @@ \subsection{Definition and universal property}
     \[ h(m \otimes n) = f(m, n) \]
     Note that the elements \( \qty{m \otimes n} \) generate \( M \otimes N \) as an \( R \)-module, so there is at most one \( h \).
     We now show that the definition of \( h \) on the pure tensors \( m \otimes n \) extends to an \( R \)-linear map \( M \otimes N \to L \).
-    The map \( R^{\oplus(M \otimes N)} \to L \) given by \( (m, n) \mapsto f(m, n) \) exists by the universal property of the direct sum.
+    The map \( R^{\oplus(M \times N)} \to L \) given by \( (m, n) \mapsto f(m, n) \) exists by the universal property of the direct sum.
     However, this map vanishes on the generators of \( K \), so it factors through the quotient \( \faktor{\mathcal F}{K} \) as required.
 \end{proof}
 The universal property given above characterises the tensor product up to isomorphism.
@@ -185,6 +185,7 @@ \subsection{Zero tensors}
 \end{example}
 
 \subsection{Monoidal structure}
+We will prove a number of elementary propositions in detail to show how tensor products are used in practice.
 \begin{proposition}[commutativity]
     There is an isomorphism \( M \otimes N \simeq N \otimes N \) mapping a pure tensor \( m \otimes n \) to \( n \otimes m \).
 \end{proposition}
@@ -579,7 +580,7 @@ \subsection{Restriction and extension of scalars}
 \( h_s \) is \( R \)-linear by the universal property.
 Defining \( \varphi : S \to \End(M \otimes_R N) \) by \( \varphi(s) = h_s \), one can check that \( h_s \) is a well-defined endomorphism and that \( \varphi \) is a ring homomorphism.
 \begin{example}
-    \( S \otimes_R R \simeq R \) as \( R \)-modules, by \( s \otimes r \mapsto s \cdot f(r) \).
+    \( S \otimes_R R \simeq S \) as \( R \)-modules, by \( s \otimes r \mapsto s \cdot f(r) \).
     This is also \( S \)-linear, since
     \[ s'(s \otimes r) = (s's \otimes r) \mapsto s's \cdot f(r) = s'(s \cdot f(r)) \]
     For example, \( \mathbb C \otimes_{\mathbb R} \mathbb R \simeq \mathbb C \) as \( \mathbb C \)-modules.
@@ -675,7 +676,7 @@ \subsection{Restriction and extension of scalars}
 \subsection{Extension of scalars on morphisms}
 Let \( f : N \to N' \) be an \( R \)-linear map, and \( M \) be an \( S \)-module.
 Then the map
-\[ \id_M \otimes f : M \otimes_R \to M \otimes_R N' \]
+\[ \id_M \otimes f : M \otimes_R N \to M \otimes_R N' \]
 is \( S \)-linear.
 Indeed,
 \[ (\id_M \otimes f)(s(m \otimes n)) = \id_M sm \otimes f(n) = s(m \otimes f(n)) = s((\id_M \otimes f)(m \otimes n)) \]
@@ -1060,14 +1061,11 @@ \subsection{Flat modules}
     \end{enumerate}
 \end{proposition}
 Note that a map \( f : M \to N \) is injective exactly when the sequence
-% https://q.uiver.app/#q=WzAsNCxbMCwwLCJNIFxcb3RpbWVzX1IgUiJdLFsxLDAsIk0gXFxvdGltZXNfUiBSIl0sWzEsMSwiTSJdLFswLDEsIk0iXSxbMCwxLCJcXGlkX00gXFxvdGltZXMgZiJdLFsxLDIsIlxcc2ltZXEiXSxbMCwzLCJcXHNpbWVxIiwyXSxbMywyLCJtIFxcbWFwc3RvIHJfMCBtIiwyXV0=
+% https://q.uiver.app/#q=WzAsMyxbMCwwLCIwIl0sWzEsMCwiTSJdLFsyLDAsIk4iXSxbMCwxXSxbMSwyLCJmIl1d
 \[\begin{tikzcd}
-	{M \otimes_R R} & {M \otimes_R R} \\
-	M & M
-	\arrow["{\id_M \otimes f}", from=1-1, to=1-2]
-	\arrow["\simeq", from=1-2, to=2-2]
-	\arrow["\simeq"', from=1-1, to=2-1]
-	\arrow["{m \mapsto r_0 m}"', from=2-1, to=2-2]
+	0 & M & N
+	\arrow[from=1-1, to=1-2]
+	\arrow["f", from=1-2, to=1-3]
 \end{tikzcd}\]
 is exact, so all of these conditions relate exact sequences.
 \begin{proof}
@@ -1115,11 +1113,37 @@ \subsection{Flat modules}
     \end{align*}
 
     \emph{(iii) implies (ii).}
-    \( T_M \) is right exact.
-    % write in more detail. need to use flatness to do the LHS of the sequence.
+    Suppose the sequence
+    % https://q.uiver.app/#q=WzAsNSxbMSwwLCJBIl0sWzIsMCwiQiJdLFszLDAsIkMiXSxbMCwwLCIwIl0sWzQsMCwiMCJdLFswLDEsImYiXSxbMSwyLCJnIl0sWzMsMF0sWzIsNF1d
+\[\begin{tikzcd}
+	0 & A & B & C & 0
+	\arrow[from=1-1, to=1-2]
+	\arrow["f", from=1-2, to=1-3]
+	\arrow["g", from=1-3, to=1-4]
+	\arrow[from=1-4, to=1-5]
+\end{tikzcd}\]
+    is exact.
+    As \( T_M \) is right exact, we obtain the exact sequence
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJNIFxcb3RpbWVzX1IgQSJdLFsxLDAsIk0gXFxvdGltZXNfUiBCIl0sWzIsMCwiTSBcXG90aW1lc19SIEMiXSxbMywwLCIwIl0sWzAsMSwiXFxpZF9NIFxcb3RpbWVzIGYiXSxbMSwyLCJcXGlkX00gXFxvdGltZXMgZyJdLFsyLDNdXQ==
+\[\begin{tikzcd}
+	{M \otimes_R A} & {M \otimes_R B} & {M \otimes_R C} & 0
+	\arrow["{\id_M \otimes f}", from=1-1, to=1-2]
+	\arrow["{\id_M \otimes g}", from=1-2, to=1-3]
+	\arrow[from=1-3, to=1-4]
+\end{tikzcd}\]
+    It suffices to show that \( \id_M \otimes f \) is injective, but this is precisely the hypothesis of (iii).
 
     \emph{(iv) implies (iii).}
-    % todo: use prop above about being zero in finitely generated submodules
+    Let \( f : N \to N' \) be \( R \)-linear and injective.
+    Let \( \sum m_i \otimes n_i \in M \otimes_R N \) be such that
+    \[ 0 = (\id_M \otimes f)\qty(\sum m_i \otimes n_i) \in M \otimes N' \]
+    Then there are finitely generated submodules \( L, L' \) of \( N, N' \) such that the \( n_i \) are elements of \( L \) and
+    \[ 0 = (\id_M \otimes f)\qty(\sum m_i \otimes n_i) \in M \otimes L' \]
+    By (iv), we obtain
+    \[ 0 = \sum m_i \otimes n_i \in M \otimes L \]
+    But \( L \) is a submodule of \( N \), so
+    \[ 0 = \sum m_i \otimes n_i \in M \otimes N \]
+    Hence \( \id_M \otimes f : M \otimes_R N \to M \otimes_R N' \) is injective.
 \end{proof}
 \begin{proposition}
     Let \( f : R \to S \) be a ring homomorphism, and let \( M \) be a flat \( R \)-module.
@@ -1156,7 +1180,7 @@ \subsection{Flat modules}
 \end{example}
 \begin{example}
     Let \( V \) be a vector space over \( \mathbb Q \).
-    Then \( \mathbb Q \otimes_{\mathbb Q} V \cong V \) as \( \mathbb Q \)-modules, given by the map \( x \otimes v \mapsto xv \).
+    Then \( \mathbb Q \otimes_{\mathbb Q} V \simeq V \) as \( \mathbb Q \)-modules, given by the map \( x \otimes v \mapsto xv \).
     However, \( \mathbb Q \otimes_{\mathbb Z} V \) is also isomorphic to \( V \), given by the same map.
     First, note that every tensor in \( \mathbb Q \otimes_{\mathbb Z} V \) is pure.
     \[ \sum \frac{a_i}{b_i} \otimes v_i = \sum \frac{1}{b_i} \otimes a_i v_i = \sum \frac{1}{b_i} \otimes b_i \frac{a_i}{b_i} v_i = \sum 1 \otimes \frac{a_i}{b_i} v_i = 1 \otimes \sum \frac{a_i}{b_i} v_i \]

iii/forcing/03_forcing.tex

@@ -78,7 +78,10 @@ \subsection{Forcing posets}
     \end{enumerate}
 \end{definition}
 \begin{proposition}
-    If \( (\mathbb P, \leq) \) is a separative preorder, it is a partial order.
+    \begin{enumerate}
+        \item If \( (\mathbb P, \leq) \) is a separative preorder, it is a partial order.
+        \item If \( (\mathbb P, \leq) \) is a poset, then it is separative if and only if whenever \( q \nleq p \), there is \( r \leq q \) such that \( r \perp p \).
+    \end{enumerate}
 \end{proposition}
 \begin{proposition}
     Suppose that \( (\mathbb P, \leq) \) is a preorder.
@@ -836,8 +839,8 @@ \subsection{\texorpdfstring{\( \mathsf{ZF} \)}{ZF} in forcing extensions}
     By the forcing theorem, there is \( p \in G \) such that \( (p \Vdash \varphi(\dot x, \dot z, \dot v))^M \).
     Hence \( \langle p, \dot x \rangle \in C \).
     So we can fix \( \dot y \in B \) such that \( (p \Vdash \varphi(\dot x, \dot y, \dot v))^M \).
-    Therefore, \( \langle \Bbbone 1, \dot y \rangle \in \dot b \).
-    Since \( \Bbbone 1 \in G \), \( \dot y^G \in \dot b^G \).
+    Therefore, \( \langle \Bbbone, \dot y \rangle \in \dot b \).
+    Since \( \Bbbone \in G \), \( \dot y^G \in \dot b^G \).
     By the forcing theorem again,
     \[ M[G] \vDash \dot y^G \in \dot b^G \wedge \varphi(\dot x^G, \dot y^G, v) \]
     Hence, collection holds.

iii/forcing/04_forcing_and_independence.tex

@@ -400,7 +400,7 @@ \subsection{Larger chain conditions}
 \end{lemma}
 \begin{theorem}
     Let \( \mathbb P \in M \) be a forcing poset, let \( (\kappa \text{ is regular})^M \), let \( (\mathbb P \) has the \( \kappa \)-chain condition\( )^M \).
-    Then \( \mathbb P \) preserves cofinalities above \( \kappa \), and hence cardinals at least \( \kappa \).
+    Then \( \mathbb P \) preserves cofinalities above \( \kappa \), and hence cardinals above \( \kappa \).
 \end{theorem}
 On the example sheet, we show that for any infinite cardinal \( \kappa \), \( \Fn_\kappa(I, J) \) has the \( (\abs{J}^{<\kappa})^+ \)-chain condition.
 In particular, \( \Fn_\kappa(\lambda \times \kappa, 2) \) has the \( (2^{<\kappa})^+ \)-chain condition.
@@ -466,7 +466,7 @@ \subsection{Closure and distributivity}
     If \( \kappa \) is regular in \( M \), then \( \Fn_\kappa(I, J)^M \) is \( <\kappa \)-closed.
 \end{lemma}
 \begin{theorem}
-    Let \( A, B, \mathbb P \in M \), let \( \kappa \) be a cardinal in \( M \) with \( (\abs{A} < \kappa)^M \), and suppose \( \mathbb P \) is \( \kappa \)-distributive in \( M \).
+    Let \( A, B, \mathbb P \in M \), let \( \kappa \) be a cardinal in \( M \) with \( (\abs{A} < \kappa)^M \), and suppose \( \mathbb P \) is \( <\kappa \)-distributive in \( M \).
     Let \( G \) be \( \mathbb P \)-generic.
     Then if \( f \in M[G] \) with \( f : A \to B \), then \( f \in M \).
 \end{theorem}
@@ -501,7 +501,7 @@ \subsection{Closure and distributivity}
     Suppose that \( \beta \) is singular in \( M[G] \).
     Fix \( \delta < \beta \) and a cofinal map \( f : \delta \to \beta \) in \( M[G] \).
     Note that \( \delta \in M \).
-    Since \( \mathbb P \) is \( \kappa \)-closed, it is \( \kappa \)-distributive, so \( f \in M \), contradicting the assumption that \( \beta \) is regular in \( M \).
+    Since \( \mathbb P \) is \( <\kappa \)-closed, it is \( <\kappa \)-distributive, so \( f \in M \), contradicting the assumption that \( \beta \) is regular in \( M \).
 \end{proof}
 \begin{theorem}
     Let \( \kappa, \lambda \) be cardinals in \( M \) such that \( \aleph_0 \leq \kappa \leq \lambda \).
@@ -744,10 +744,10 @@ \subsection{Product forcing}
     For the other direction, \( G \in M[G \times H] \) and \( M \subseteq M[G \times H] \) so \( M[G] \subseteq M[G \times H] \), but also \( H \in M[G \times H] \) so \( M[G][H] \subseteq M[G \times H] \).
 \end{proof}
 Recall that we started with a model of \( \mathsf{ZFC} + \mathsf{GCH} \) and forced with
-\[ G_0 \text{ is } \Fn(\omega_3 \times \omega, 2)^M \text{-generic};\quad G_1 \text{ is } \Fn_(\omega_5 \times \omega_1, 2)^{M[G_0]} \text{-generic} \]
+\[ G_0 \text{ is } \Fn(\omega_3 \times \omega, 2)^M \text{-generic};\quad G_1 \text{ is } \Fn(\omega_5 \times \omega_1, 2)^{M[G_0]} \text{-generic} \]
 and found that \( M[G_0][G_1] \vDash \mathsf{CH} \).
 But if instead we used
-\[ G_0 \text{ is } \mathbb P_0 = \Fn(\omega_5 \times \omega_1, 2)^M \text{-generic};\quad G_1 \text{ is } \mathbb P_1 = \Fn_(\omega_3 \times \omega, 2)^{M[G_0]} \text{-generic} \]
+\[ G_0 \text{ is } \mathbb P_0 = \Fn(\omega_5 \times \omega_1, 2)^M \text{-generic};\quad G_1 \text{ is } \mathbb P_1 = \Fn(\omega_3 \times \omega, 2)^{M[G_0]} \text{-generic} \]
 then we obtain \( M[G_0][G_1] \vDash 2^{\aleph_0} = \aleph_3 + 2^{\aleph_1} = \aleph_5 \).
 However, \( \mathbb P_0 \) is \( <\omega_1 \)-closed, so does not add new sequences of length \( \omega \).
 Thus \( \mathbb P_1 = \Fn(\omega_3 \times \omega, 2)^M \).

iii/mtncl/03_ultraproducts.tex

@@ -181,14 +181,13 @@ \subsection{\L{}o\'s' theorem}
 \end{corollary}
 \begin{proof}
     If \( \mathcal T \) is finite, the result is trivial, so we may suppose it is infinite.
-    Let \( I \) be the set of all finite subtheories of \( \mathcal T \).
-    For each axiom \( \varphi \in \mathcal T \), define \( X_\varphi \) to be the set of finite subtheories of \( \mathcal T \) that include \( \varphi \) as an axiom, and let
-    \[ D = \qty{Y \subseteq I \mid \exists \varphi \in \mathcal T.\, X_\varphi \subseteq Y} \]
-    Then \( D \) is a proper filter on \( Y \), so by the ultrafilter principle, it can be extended to an ultrafilter \( \mathcal U \).
+    Let \( I \) be the set of all finite subtheories of \( \mathcal T \), and let
+    % For each axiom \( \varphi \in \mathcal T \), define \( X_\varphi \) to be the set of finite subtheories of \( \mathcal T \) that include \( \varphi \) as an axiom, and let
+    \[ D = \qty{Y \subseteq I \mid \exists \Delta \in I.\, \forall X \in Y.\, \Delta \subseteq X} \]
+    Then \( D \) is a proper filter on \( I \), so by the ultrafilter principle, it can be extended to an ultrafilter \( \mathcal U \).
     Using the axiom of choice, let \( \mathcal M_\Delta \) be a model of \( \Delta \) for each finite subtheory \( \Delta \in I \).
-    Then, for any \( \varphi \in \mathcal T \),
-    \[ X_\varphi \subseteq \qty{\Delta \in I \mid M_\Delta \vDash \varphi} \]
-    Thus \( \qty{\Delta \in I \mid M_\Delta \vDash \varphi} \in D \subseteq \mathcal U \).
+    Then, for any \( \varphi \in \mathcal T \), we have
+    \[ \{ Y \subseteq I \mid \forall X \in Y.\, \varphi \in X \} \in D \subseteq \mathcal U \]
     Then by \L{}o\'s' theorem, the ultraproduct \( \faktor{\prod_{\Delta \in I} \mathcal M_\Delta}{\mathcal U} \) models \( \varphi \).
     In particular, the ultraproduct models \( \mathcal T \).
 \end{proof}

iii/mtncl/04_types.tex

@@ -3,7 +3,7 @@ \subsection{Definitions}
     Let \( X \subseteq \mathcal M^n \) be a subset of an \( \mathcal L \)-structure \( \mathcal M \), and let \( P \subseteq \mathcal M \).
     We say that \( X \) is \emph{definable} in \( \mathcal L \) with \emph{parameters} in \( P \) if there is a tuple \( \vb p \in P \) and an \( \mathcal L_P \)-formula \( \varphi(\vb x, \vb y) \) such that
     \[ X = \varphi(\vb x, \vb p) = \qty{\vb m \in \mathcal M^n \mid \mathcal M \vDash \varphi(\vb m, \vb p)} \]
-    If \( P = M \), we say that \( X \) is \emph{definable}.
+    If \( P = \mathcal M \), we say that \( X \) is \emph{definable}.
 \end{definition}
 \begin{example}
     Consider the usual natural numbers as a structure for the language generated by the signature \( (+, \cdot, 0, 1) \).
@@ -106,7 +106,7 @@ \subsection{Stone spaces}
 \[ \lBrack\varphi\rBrack = \qty{p \in S_n^{\mathcal M}(A) \mid \varphi \in p} \]
 Note that
 \[ \lBrack\varphi \vee \psi\rBrack = \lBrack\varphi\rBrack \cup \lBrack\psi\rBrack;\quad \lBrack\varphi \wedge \psi\rBrack = \lBrack\varphi\rBrack \cap \lBrack\psi\rBrack \]
-These serve as the basic open sets for a topology on \( S_n^{\mathcal m}(A) \), so an open set is an arbitrary union of open sets of this form.
+These serve as the basic open sets for a topology on \( S_n^{\mathcal M}(A) \), so an open set is an arbitrary union of open sets of this form.
 Moreover, each of these basic open sets \( \lBrack\varphi\rBrack \) is the complement of another basic open set \( \lBrack\neg\varphi\rBrack \), so these open sets are also closed.
 The \( S_n^{\mathcal M}(A) \) are called \emph{Stone spaces}, which are compact and totally disconnected topological spaces.
 \begin{example}
@@ -161,7 +161,7 @@ \subsection{Omitting types}
     First, suppose \( s = 2i \).
     These sentences will be designed to turn \( C \) into the domain of an elementary substructure of some model of \( \mathcal T^\star \).
     Suppose that \( \varphi_i = \exists x.\, \psi(x) \) is existential, with parameters in \( C \) as \( \varphi \) is an \( \mathcal L_C \)-formula.
-    Suppose \( \mathcal T \vDash \theta_s \to \varphi_i \).
+    Suppose also that \( \mathcal T \vDash \theta_s \to \varphi_i \).
     As only finitely many constants from \( C \) have been used so far, we can find some unused \( c \in C \).
     Let
     \[ \theta_{s + 1} = \theta_s \wedge \psi(c) \]
@@ -171,7 +171,7 @@ \subsection{Omitting types}
 
     Now, suppose \( s = 2i + 1 \).
     These sentences will be designed to ensure that \( C \) omits \( p \).
-    Let \( \overline d_i = (e_1, \dots, e_n) \).
+    Let \( \vb d_i = (e_1, \dots, e_n) \).
     Remove every occurrence of the \( e_j \) from \( \theta_s \) by replacing it with the variable \( x_j \), and replace every occurrence of other constants in \( C \) with a fresh variable \( x_c \), together with a quantifier \( \exists x_c \) in front of the formula.
     This yields an \( \mathcal L \)-formula \( \psi(x_1, \dots, x_n) \).
     For example, if
@@ -182,13 +182,13 @@ \subsection{Omitting types}
     We define \( \theta_{s+1} \) in such a way that \( \vb d_i \) cannot realise \( p \).
     \[ \theta_{s + 1} = \theta_s \wedge \neg \varphi(\vb d_i) \]
     This is consistent, because there must be some \( \vb n \in \mathcal N \vDash \mathcal T \) such that
-    \[ \mathcal N \vDash \psi(\vb n) \wedge \neg \psi(\vb n) \]
+    \[ \mathcal N \vDash \psi(\vb n) \wedge \neg \varphi(\vb n) \]
     and we can turn \( \mathcal N \) into an \( \mathcal L_C \)-structure that models \( \theta_{s+1} \) by interpreting \( \vb d_i \) as \( \vb n \), and interpreting the constants in \( C \) but not in \( \vb d \) as the respective witnesses to the existential statements \( \exists x_c \) within \( \psi \).
 
 	Let \( \mathcal T^\star \) be \( \mathcal T \) together with all of the \( \theta_s \).
 	Note that each \( \mathcal T \cup \qty{\theta_s} \) is consistent, and each \( \theta_{s+1} \) implies \( \theta_s \), so by compactness, \( \mathcal T^\star \) must be consistent.
 	Moreover, if \( \mathcal M \) is a model of \( \mathcal T^\star \), the construction of \( \theta_{2i+1} \) ensures that \( C \) has a witness to \( \varphi_i \) that holds in \( \mathcal M \).
-	Thus, by that Tarski--Vaught test, \( \mathcal C \) is the domain of an elementary substructure of \( \mathcal M \).
+	Thus, by the Tarski--Vaught test, \( C \) is the domain of an elementary substructure of \( \mathcal M \).
 	If \( \vb c \in C \vDash \mathcal T^\star \), then \( \vb c = \vb d_i \) for some \( i \).
 	As \( C \vDash \theta_{2i + 2} \), we have \( \neg \varphi(\vb c) \) for some \( \varphi \) in the type \( p \).
 	Hence \( \vb c \) cannot realise the type \( p \) in \( C \).