Lots of little fixes

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

ii/nf/02_ideals.tex

@@ -128,7 +128,7 @@ \subsection{Unique factorisation of ideals}
 \begin{proof}
     Otherwise, as \( \mathcal O_K \) is Noetherian, there exists a ideal \( \mathfrak a \) which is maximal with this property.
     In particular, \( \mathfrak a \) is not prime.
-    So there exists \( x, y \in \mathcal O_K \) with \( x, y \not\in \mathfrak a \) but \( xy \in \mathfrak a \).
+    So there exists \( x, y \in \mathcal O_K \) with \( x \) or \( y \) not in \( \mathfrak a \) but \( xy \in \mathfrak a \).
     So \( \mathfrak a \subsetneq \mathfrak a + (x) \).
     But then, \( \mathfrak a + (x) \) contains a product of prime ideals \( \mathfrak p_1, \dots, \mathfrak p_r \) with \( \mathfrak p_1\dots \mathfrak p_r \subseteq \mathfrak a + (x) \).
     Similarly,  there exist prime ideals \( \mathfrak q_1, \dots \mathfrak q_s \) such that \( \mathfrak q_1 \dots\mathfrak q_s \subseteq \mathfrak a + (y) \).

iii/alggeom/02_sheaves.tex

@@ -96,7 +96,7 @@ \subsection{Stalks}
 \begin{proposition}
     Let \( f : \mathcal F \to \mathcal G \) be a morphism of sheaves on \( X \).
     Suppose that for all \( p \in X \), the induced map \( f_p : \mathcal F_p \to \mathcal G_p \) given by
-    \[ f_p((U, s)) = (U, \mathcal f_U(s)) \]
+    \[ f_p((U, s)) = (U, f_U(s)) \]
     is an isomorphism.
     Then \( f \) is an isomorphism.
 \end{proposition}

iii/alggeom/05_modules_over_the_structure_sheaf.tex

@@ -77,7 +77,7 @@ \subsection{Quasi-coherence}
     We can cover each such \( V \) by distinguished affines of the form \( U_g \) for some \( g \in A \).
     Then \( \eval{\mathcal F}_{U_g} = (M \otimes_B A_g)^{\mathrm{sh}} \), as \( \eval{F}_V \) is quasi-coherent.
     But recall that \( \Spec A \) is quasi-compact: every open cover has a finite subcover.
-    So finitely many \( U_{g_i} \) will suffice to cover \( A \) by open sets such that \( \mathcal F \) restricts to \( M_i^{\mathrm{sh}} \) on \( U_{g_i} \).
+    So finitely many \( U_{g_i} \) will suffice to cover \( X \) by open sets such that \( \mathcal F \) restricts to \( M_i^{\mathrm{sh}} \) on \( U_{g_i} \).
     Then the lemma follows from formal properties of localisation.
 \end{proof}
 We now prove the main proposition.

iii/cat/07_additive_and_abelian_categories.tex

@@ -24,7 +24,7 @@ \subsection{Additive categories}
     \end{enumerate}
 \end{lemma}
 \begin{proof}
-    In each part, as (a) and (b) are dual and (c) is self-dual, it suffices to prove the equivalence of (a) and (b).
+    In each part, as (a) and (b) are dual and (c) is self-dual, it suffices to prove the equivalence of (a) and (c).
 
     \emph{Part (i).}
     If \( A \) is terminal, then it has exactly one morphism \( A \to A \), so this must be the zero morphism.
@@ -171,7 +171,7 @@ \subsection{Kernels and cokernels}
     If \( f \cong \ker \coker f \), it is clearly normal.
     Now suppose \( f = \ker g \).
     Then \( g \) factors through the cokernel of \( f \), so \( g (\ker \coker f) = 0 \).
-    Thus \( \ker coker f \leq f \) in \( \operatorname{Sub}(B) \).
+    Thus \( \ker \coker f \leq f \) in \( \operatorname{Sub}(B) \).
     But \( (\coker f) f = 0 \), so \( f \leq \ker \coker f \), so they are isomorphic as subobjects of \( B \).
 \end{proof}
 \begin{corollary}
@@ -252,7 +252,7 @@ \subsection{Abelian categories}
     \( \begin{pmatrix}
         f \\ g
     \end{pmatrix} \) is the kernel of \( \begin{pmatrix}
-        h \\ -k
+        h & -k
     \end{pmatrix} \) if and only if
     % https://q.uiver.app/#q=WzAsMyxbMCwwLCJBIl0sWzEsMCwiQiJdLFswLDEsIkMiXSxbMCwxLCJmIl0sWzAsMiwiZyIsMl1d
 \[\begin{tikzcd}

iii/commalg/04_integrality_finiteness_finite_generation.tex

@@ -521,7 +521,7 @@ \subsection{Integrality over ideals}
 \begin{proof}
     If \( b \in B \) is integral over \( \mathfrak a \), then
     \[ b^n + a_1 b^{n-1} + \dots + a_n b^0 = 0;\quad a_i \in \mathfrak a \]
-    In particular, \( \mathfrak b \) lies in \( \overline A \), and so all of its powers lie in \( \overline A \) as \( \overline A \) is a ring.
+    In particular, \( b \) lies in \( \overline A \), and so all of its powers lie in \( \overline A \) as \( \overline A \) is a ring.
     Using the integrality equation for \( b \), we observe that \( b^n \in \mathfrak a \overline A \), hence \( b \in \sqrt{\mathfrak a \overline A} \).
 
     Now, suppose \( b \in \sqrt{\mathfrak a \overline A} \).
@@ -537,7 +537,7 @@ \subsection{Integrality over ideals}
     Thus by the Cayley--Hamilton theorem,
     \[ f^\ell + \alpha_1 f^{\ell - 1} + \dots + \alpha_\ell f^0 = 0 \in \End_R M;\quad \alpha_i \in \mathfrak a \]
     Evaluating this at \( 1_A \in M \),
-    \[ b^{m\ell} + \alpha_1 b^{m(\ell - 1)} + \dots + \alpha_\ell b^0 = 0 \in B \]
+    \[ b^{n\ell} + \alpha_1 b^{n(\ell - 1)} + \dots + \alpha_\ell b^0 = 0 \in B \]
     This is an integrality relation for \( b \) is \( \mathfrak a \)-integral.
 \end{proof}
 Hence, the integral closure of an ideal is closed under sums and products.
@@ -581,7 +581,7 @@ \subsection{Integrality over ideals}
     \[ f = \prod_{i=1}^\ell (T - \alpha_i);\quad \alpha_1 = b, \alpha_i \in \Omega \]
     We want to show that the coefficients of \( f \) are in \( \sqrt{\mathfrak a} \).
     By the previous proposition, together with the fact that \( A \) is integrally closed, the integral closure of \( \mathfrak a \) in \( FF(A) \) is \( \sqrt{\mathfrak a} \subseteq A \).
-    So it suffices to show that the coefficients of \( f \) lie in \( FF(A) \) and are integrally closed in \( A \).
+    So it suffices to show that the coefficients of \( f \) lie in \( FF(A) \) and are integral over \( \mathfrak a \).
     As \( f \) is the minimal polynomial over \( FF(A) \), the first part holds by definition.
 
     Expanding brackets in the equation for \( f \), the coefficients of \( f \) are sums of products of the \( \alpha_i \).

iii/commalg/07_dimension_theory.tex

@@ -266,7 +266,7 @@ \subsection{Dimension theory of local Noetherian rings}
 \end{proof}
 \begin{corollary}[Krull's height theorem]
     Let \( A \) be a Noetherian ring, and let \( \mathfrak a = (x_1, \dots, x_r) \) be an ideal of \( A \).
-    Let \( \mathfrak p \) be a minimal prime ideal of \( A \).
+    Let \( \mathfrak p \) be a minimal prime ideal of \( \mathfrak a \).
     Then \( \operatorname{ht}(\mathfrak p) \leq r \).
 \end{corollary}
 \begin{proof}

iii/forcing/01_set_theory.tex

@@ -230,7 +230,7 @@ \subsection{The L\'evy hierarchy}
 \end{definition}
 \begin{example}
     The formula \( \forall v_1.\, \exists v_2.\, \forall v_3.\, (v_4 = v_3) \) is \( \Pi_3 \).
-    But \( \forall v_1.\, (v_1 = v_2) \wedge v_3 = v_4 \) is not \( \Pi_n \) or \( \Sigma_n \) for any \( n \).
+    But \( (\forall v_1.\, v_1 = v_2) \wedge v_3 = v_4 \) is not \( \Pi_n \) or \( \Sigma_n \) for any \( n \).
 \end{example}
 \begin{definition}
     Given an \( \mathcal L_\in \)-theory \( T \), let \( \Sigma_n^T \) be the class of formulas \( \Gamma \) such that for any \( \varphi \in \Gamma \), there exists \( \psi \in \Sigma_n \) such that \( T \vdash \varphi \leftrightarrow \psi \).

iii/forcing/02_constructibility.tex

@@ -13,7 +13,7 @@ \subsection{Definable sets}
 \begin{remark}
     \begin{enumerate}
         \item \( M \in \operatorname{Def}(M) \).
-        \item \( M \subseteq \operatorname{Def}(M) \subseteq \mathcal \mathcal P(M) \).
+        \item \( M \subseteq \operatorname{Def}(M) \subseteq \mathcal P(M) \).
     \end{enumerate}
 \end{remark}
 This definition involves a quantification over infinitely many formulas, so is not yet fully formalised.
@@ -53,7 +53,7 @@ \subsection{Defining the constructible universe}
     By the previous lemma, it suffices to check that \( \mathsf{ZF}^{\mathrm{L}} \) holds.
     \begin{itemize}
         \item Since \( \mathrm{L} \) is transitive, \( \mathrm{L} \) satisfies extensionality and foundation.
-        \item For the axiom of empty set, we use the fact that \( \varnothing^{\mathrm{L}} = \varnothing = \mathrm{L}_0 = \mathrm{L} \).
+        \item For the axiom of empty set, we use the fact that \( \varnothing^{\mathrm{L}} = \varnothing = \mathrm{L}_0 \in \mathrm{L} \).
         \item For pairing, given \( a, b \in \mathrm{L} \), we must show \( \qty{a, b} \in \mathrm{L} \).
         Fix \( \alpha \) such that \( a, b \in \mathrm{L}_\alpha \).
         Then
@@ -99,10 +99,10 @@ \subsection{G\"odel functions}
         \item \( \mathcal F_3(x, y) = x \setminus y \);
         \item \( \mathcal F_4(x, y) = x \times y \);
         \item \( \mathcal F_5(x, y) = \dom x = \qty{\pi_1(z) \mid z \in x \wedge z \text{ is an ordered pair}} \);
-        \item \( \mathcal F_6(x, y) = \ran x \qty{\pi_2(z) \mid z \in x \wedge z \text{ is an ordered pair}} \);
+        \item \( \mathcal F_6(x, y) = \ran x = \qty{\pi_2(z) \mid z \in x \wedge z \text{ is an ordered pair}} \);
         \item \( \mathcal F_7(x, y) = \qty{\langle u, v, w \rangle \mid \langle u, v \rangle \in x, w \in y} \);
         \item \( \mathcal F_8(x, y) = \qty{\langle u, w, v \rangle \mid \langle u, v \rangle \in x, w \in y} \);
-        \item \( \mathcal F_9(x, y) = \qty{\langle v, u \rangle \in y \in x \mid u = v} \);
+        \item \( \mathcal F_9(x, y) = \qty{\langle v, u \rangle \in y \times x \mid u = v} \);
         \item \( \mathcal F_{10}(x, y) = \qty{\langle v, u \rangle \in y \times x \mid u \in v} \).
     \end{enumerate}
 \end{definition}
@@ -415,7 +415,7 @@ \subsection{Well-ordering the universe}
         Suppose \( x, y \notin \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \).
         We say \( x <_{\alpha + 1}^{n+1} \) if either
         \begin{enumerate}
-            \item the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( \mathcal y = \mathcal F_i(u, v) \); or
+            \item the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( y = \mathcal F_i(u, v) \); or
             \item these indices \( i \) are equal, and the \( <_{\alpha + 1}^n \)-least \( u \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) such that there exists \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the \( <_{\alpha + 1}^n \)-least \( u \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) such that there exists \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( y = \mathcal F_i(u, v) \); or
             \item both of these coincide, and \( <_{\alpha + 1}^n \)-least \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the least \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( y = \mathcal F_i(u, v) \).
         \end{enumerate}
@@ -527,7 +527,7 @@ \subsection{The generalised continuum hypothesis in \texorpdfstring{\( \mathrm{L
 \subsection{Combinatorial properties}
 \begin{definition}
     Let \( \Omega \) be either a regular cardinal or the class of all ordinals.
-    A subclass \( C \subseteq \Omega \) is said to be a \emph{club} or \emph{closed and unbounded} if it is
+    A subclass \( C \subseteq \Omega \) is said to be a \emph{club}, or \emph{closed and unbounded}, if it is
     \begin{enumerate}
         \item \emph{closed}: for all \( \gamma \in \Omega \), we have \( \sup(C \cap \gamma) \in C \);
         \item \emph{unbounded}: for all \( \alpha \in \Omega \) there exists \( \beta \in C \) with \( \beta > \alpha \).
@@ -562,7 +562,7 @@ \subsection{Combinatorial properties}
     Then the \emph{square principle} \( \mdwhtsquare_\kappa \) is the assertion that there exists a sequence \( (C_\alpha) \) indexed by the limit ordinals \( \alpha \) in \( \kappa^+ \), such that
     \begin{enumerate}
         \item \( C_\alpha \) is a club subset of \( \alpha \);
-        \item if \( \beta \) is a limit ordinal of \( C_\alpha \) then \( C_\beta = C_\alpha \cap \beta \); then
+        \item if \( \beta \) is a limit ordinal of \( C_\alpha \) then \( C_\beta = C_\alpha \cap \beta \); and
         \item if \( \cf(\alpha) < \kappa \) then \( \abs{C_\alpha} < \kappa \).
     \end{enumerate}
 \end{definition}

iii/forcing/03_forcing.tex

@@ -69,7 +69,7 @@ \subsection{Forcing posets}
     This is called \emph{Jerusalem notation}.
 \end{remark}
 The notation \( \mathbb P \in M \) abbreviates \( (\mathbb P, \leq_{\mathbb P}, \Bbbone_{\mathbb P}) \in M \).
-Note that by transitivity if \( \mathbb P \) is an element of \( M \), then \( \Bbbone_{\mathbb P} \in M \), but we do not necessarily have \( \leq_{\mathbb P} \in M \).
+Note that by transitivity if \( \mathbb P \) is an element of \( M \), then \( \Bbbone_{\mathbb P} \in M \), but we do not necessarily have \( {\leq_{\mathbb P}} \in M \).
 \begin{definition}
     A preorder is \emph{separative} if whenever \( p \neq q \), exactly one of the following two cases holds:
     \begin{enumerate}

iii/gc/01_definitions_and_resolutions.tex

@@ -82,7 +82,7 @@ \subsection{???}
 \end{definition}
 We say that \( M_\bullet \) is \emph{exact} at \( M_n \) if \( \im d_{n+1} = \ker d_n \), and we say it is \emph{exact} if it is exact at all \( M_n \) for \( t < n < s \).
 The \emph{homology} of this chain complex is
-\[ H_s(M_\bullet) = \ker d_s;\quad H_n(M_\bullet) = \faktor{\ker d_n}{\im d_{n+1}};\quad H_t(M_\bullet) = \coker d_{t-1} = \faktor{M_t}{\im d_{t-1}} \]
+\[ H_s(M_\bullet) = \ker d_s;\quad H_n(M_\bullet) = \faktor{\ker d_n}{\im d_{n+1}};\quad H_t(M_\bullet) = \coker d_{t-1} = \faktor{M_t}{\im d_{t+1}} \]
 A \emph{short exact sequence} is an exact chain complex of the form
 % https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzEsMCwiTV8xIl0sWzIsMCwiTV8yIl0sWzMsMCwiTV8zIl0sWzQsMCwiMCJdLFswLDFdLFsxLDIsIlxcYWxwaGEiXSxbMiwzLCJcXGJldGEiXSxbMyw0XV0=
 \[\begin{tikzcd}
@@ -274,7 +274,7 @@ \subsection{???}
         &+ (-1)^{n-1} [g_1 | \dots | g_{n-1} g_n] \\
         &+ (-1)^n [g_1 | \dots | g_{n-1}]
     \end{align*}
-    One can verify explicitly that there are chain maps as required, giving a free resolution
+    One can verify explicitly that these are chain maps as required, giving a free resolution
     % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGNkb3RzIl0sWzEsMCwiRl8xIl0sWzIsMCwiRl8wIl0sWzMsMCwiXFxtYXRoYmIgWiJdLFswLDFdLFsxLDJdLFsyLDNdXQ==
 \[\begin{tikzcd}
 	\cdots & {F_1} & {F_0} & {\mathbb Z}

iii/gc/02_low_degree_cohomology.tex

@@ -29,22 +29,22 @@ \subsection{Degree 1}
 Then \( M \cong \qty{(m, 1) \mid m \in M} \) is a normal subgroup of \( M \rtimes G \).
 Also, \( G \cong \qty{(0, g) \mid g \in G} \), and conjugation by \( \qty{(0, g) \mid g \in G} \) corresponds to the \( G \)-action on the module \( M \).
 Further,
-\[ \faktor{M \rtimes G}{\qty{(0, g) \mid g \in G}} \cong G \]
+\[ \faktor{M \rtimes G}{\qty{(m, 1) \mid m \in M}} \cong G \]
 There is a group homomorphism \( s : G \to M \rtimes G \) given by \( g \mapsto (0, g) \), such that \( \pi_2 \circ s = \id \) where \( \pi_2 \) is the second projection.
 Such a map \( s \) is called a \emph{splitting}.
 Given another splitting \( s_1 : G \to M \rtimes G \) such that \( \pi_2 \circ s_1 = \id \), we define \( \psi_{s_1} : G \to M \) by
 \[ s_1(g) = (\psi_{s_1}(g), g) \in M \rtimes G \]
 Then \( \psi_{s_1} \) is a 1-cocycle.
 Given two splittings \( s_1, s_2 \), the difference \( \psi_{s_1} - \psi_{s_2} \) is a coboundary precisely when there exists \( m \) such that \( (m, 1)s_1(g)(m,1)^{-1} = s_2(g) \).
-Conversely, a 1-cocycle \( \varphi \in Z^1(G, M) \), there is a splitting \( s_1 : G \to M \rtimes G \) such that \( \varphi = \psi_{s_1} \).
+Conversely, given a 1-cocycle \( \varphi \in Z^1(G, M) \), there is a splitting \( s_1 : G \to M \rtimes G \) such that \( \varphi = \psi_{s_1} \).
 \begin{theorem}
     \( H^1(G, M) \) bijects with the \( M \)-conjugacy classes of splittings.
 \end{theorem}
 
 \subsection{Degree 2}
 \begin{definition}
     Let \( G \) be a group and \( M \) be a \( \mathbb Z G \)-module.
-    An \emph{extension} of \( G \) by \( M \) is a group \( E \) with a sequence of group homomorphisms
+    An \emph{extension} of \( G \) by \( M \) is a group \( E \) with an exact sequence of group homomorphisms
     % https://q.uiver.app/#q=WzAsNSxbMCwwLCIwIl0sWzEsMCwiTSJdLFsyLDAsIkUiXSxbMywwLCJHIl0sWzQsMCwiMSJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdXQ==
 \[\begin{tikzcd}
 	0 & M & E & G & 1
@@ -53,7 +53,6 @@ \subsection{Degree 2}
 	\arrow[from=1-3, to=1-4]
 	\arrow[from=1-4, to=1-5]
 \end{tikzcd}\]
-    where the maps are group homomorphisms.
     \( M \) embeds into \( E \), so its image (also called \( M \)) is an abelian normal subgroup of \( E \).
     This is acted on by conjugation by \( E \), and so we obtain an induced action of \( \faktor{E}{M} \cong G \), which must match the given \( G \)-action on \( M \).
 \end{definition}

iii/mtncl/05_indiscernibles.tex

@@ -112,5 +112,5 @@ \subsection{???}
 	This in particular implies that \( \operatorname{Th}_{\mathcal L}(\mathcal N^\star) \) is Skolem, as \( \operatorname{Th}(\mathcal M) \) is Skolem and \( \operatorname{Th}(\mathcal M) \subseteq \operatorname{Th}(\mathcal M, \eta) \).
 	It then follows that \( \mathcal S \) is an elementary substructure of \( \mathcal N^\star \), and is generated by \( \omega \).
 	Then, \( \operatorname{Th}(\mathcal M, \eta) \subseteq \operatorname{Th}(\mathcal S, \omega) \).
-	Finally, sentences in \( \mathcal T \) ensure that \( \omega \) is indiscernible in \( S \) by construction, so the stretching lemma gives an Ehrenfeucht--Mostowski functor \( F \) with \( S = F(\omega) \), which completes the proof by the previous lemma.
+	Finally, sentences in \( \mathcal T \) ensure that \( \omega \) is indiscernible in \( \mathcal S \) by construction, so the stretching lemma gives an Ehrenfeucht--Mostowski functor \( F \) with \( \mathcal S = F(\omega) \), which completes the proof by the previous lemma.
 \end{proof}