Lectures 07B

iii/alggeom/01_introduction.tex

@@ -200,7 +200,7 @@ \subsection{Spectrum of a ring}
     In general, points are not closed.
 \end{example}
 
-\subsection{Distinguished opens and localisations}
+\subsection{Distinguished opens and localisation}
 \begin{definition}
     Let \( f \in A \).
     Define the \emph{distinguished open} corresponding to \( f \) to be

iii/alggeom/02_sheaves.tex

@@ -310,3 +310,47 @@ \subsection{Kernels and cokernels}
     We say that \( \mathcal F \) is a \emph{subsheaf} of \( \mathcal G \), written \( \mathcal F \subseteq \mathcal G \), if there are inclusions \( \mathcal F(U) \subseteq \mathcal G(U) \) compatible with the restriction maps.
 \end{definition}
 Kernels are examples of subsheaves.
+
+\subsection{Moving between spaces}
+Let \( f : X \to Y \) be a continuous map of topological spaces, and let \( \mathcal F \) and \( \mathcal G \) be sheaves on \( X \) and \( Y \) respectively.
+\begin{definition}
+    The presheaf \emph{pushforward} or \emph{direct image} \( f_\star \mathcal F \) is the presheaf on \( Y \) given by
+    \[ f_\star\mathcal F(U) = \mathcal F(f^{-1}(U)) \]
+\end{definition}
+\begin{proposition}
+    The presheaf pushforward of a sheaf is a sheaf.
+\end{proposition}
+\begin{proof}
+    % Trivial exercise
+\end{proof}
+\begin{definition}
+    The \emph{inverse image presheaf} \( (f^{-1} \mathcal G)^{\mathrm{pre}} \) is the presheaf on \( X \) given by
+    \[ (f^{-1}\mathcal G)^{\mathrm{pre}}(V) = \faktor{\qty{(s_U, U) \mid f(V) \subseteq U, s_U \in \mathcal G(U)}}{\sim} \]
+    where \( \sim \) identifies pairs that agree on a smaller open set containing \( f(V) \).
+    The \emph{inverse image sheaf} is \( f^{-1} \mathcal G = ((f^{-1} \mathcal G)^{\mathrm{pre}})^{\mathrm{sh}} \).
+\end{definition}
+% the ~ stuff are all examples of colimits
+\begin{example}
+    The inverse image presheaf need not be a sheaf, even when \( f \) is an open map.
+    Let \( Y \) be a topological space, and let \( X = Y \sqcup Y \).
+    Take \( \mathcal G = \underline{\mathbb Z} \) the constant sheaf, and \( \mathcal F = (f^{-1} \mathcal G)^{\mathrm{pre}} \).
+    Let \( U \subseteq Y \) be open, and let \( V = f^{-1}(U) \).
+    Then \( \mathcal F(V) = \mathcal G(U) = \mathbb Z \), assuming \( U \) is connected.
+    But \( V = U \sqcup U \), so \( \mathcal F^{\mathrm{sh}}(V) = \mathcal G(U) \times \mathcal G(U) = \mathbb Z^2 \).
+\end{example}
+\begin{example}
+    Let \( \mathcal F \) be a sheaf on \( X \), and let \( \pi \) be the map from \( X \) to a point.
+    Then \( f_\star \mathcal F \) is a sheaf on a point, which is just an abelian group, specifically \( \mathcal F(\pi^{-1}(\qty{\bullet})) = \mathcal F(X) \).
+\end{example}
+We will use the notation
+\[ \mathcal F(X) = \Gamma(X, \mathcal F) = H^0(X, \mathcal F) \]
+where \( \Gamma \) is called the \emph{global sections}, and \( H_0 \) is called the \emph{0th cohomology} with coefficients in \( \mathcal F \).
+
+For \( p \in X \), \( i : \qty{p} \to X \).
+Let \( \mathcal G \) be a sheaf on \( p \), which is an abelian group \( A \).
+Consider the sheaf \( i_\star \mathcal G \) on \( X \), defined by
+\[ (i_\star \mathcal G)(U) = \begin{cases}
+    0 & \text{if } p \notin U \\
+    A & \text{if } p \in U
+\end{cases} \]
+This is called the \emph{skyscraper} at \( p \) with value \( A \).

iii/alggeom/03_schemes.tex

@@ -0,0 +1,47 @@
+We will now use the notation \( \eval{f}_U \) for \( \res^V_U f \).
+% Idea: Spec A has a sheaf O_{Spec A} such that value at U_f is A_f; globalise this to get the notion of a scheme.
+
+\subsection{Localisation}
+\begin{definition}
+    Let \( A \) be a ring and \( S \subseteq A \) be a multiplicatively closed set.
+    The \emph{localisation} of \( A \) at \( S \) is
+    \[ S^{-1}A = \faktor{\qty{(a, s) \mid a \in A, s \in S}}{\sim} \]
+    where
+    \[ (a, s) \sim (a', s') \iff \exists s'' \in S,\, s''(as' - a's) = 0 \in A \]
+\end{definition}
+Examples of multiplicatively closed sets include the set of powers of a fixed element, or the complement of a prime ideal.
+The pair \( (a, s) \) represents \( \frac{a}{s} \).
+The extra \( s'' \) term represents a unit in this new ring, which may be needed in rings that are not integral domains.
+\begin{remark}
+    The natural map \( A \to S^{-1} A \) need not be injective, for example, if \( S \) contains a zero divisor.
+\end{remark}
+We aim to define a sheaf \( \mathcal O_{\Spec A} \) on the topological space \( \Spec A \), such that the stalk at a prime \( \mathfrak p \) is \( (A \setminus \mathfrak p)^{-1} A \), and if \( U_f \) is a distinguished open, then \( \mathcal O_{\Spec A}(U_f) = A_f \).
+
+\subsection{Sheaves on a base}
+\begin{definition}
+    Let \( X \) be a topological space and \( \mathcal B \) be a basis for the topology.
+    A \emph{sheaf on the base \( \mathcal B \)} consists of assignments \( B_i \mapsto F(B_i) \) of abelian groups, with restriction maps \( \res_{B_j}^{B_i} : F(B_i) \to F(B_j) \) whenever \( B_j \subseteq B_i \) such that,
+    \begin{enumerate}
+        \item \( \res^{B_i}_{B_i} = \id_{B_i} \);
+        \item \( \res^{B_j}_{B_k} \circ \res^{B_i}_{B_j} = \res^{B_i}_{B_k} \)
+    \end{enumerate}
+    with the additional axioms that
+    \begin{enumerate}
+        \item if \( B = \bigcup B_i \) with \( B, B_i \in \mathcal B \) and \( f, g \in F(B) \) such that \( \eval{f}_{B_i} = \eval{g}_{B_i} \) for all \( i \), then \( f = g \);
+        \item if \( B = \bigcup B_i \) as above, with \( f_i \in F(B_i) \) such that for all \( i, j \) and \( B' \subseteq B_i \cap B_j \) with \( B' \in \mathcal B \), \( \eval{f_i}_{B'} = \eval{f_j}_{B'} \), then there exists \( f \in F(B) \) with \( \eval{f}_{B_i} = f_i \).
+    \end{enumerate}
+\end{definition}
+This is very similar to the definition of a sheaf, but only specified on the basis.
+\begin{proposition}
+    Let \( F \) be a sheaf on a base \( \mathcal B \) of \( X \).
+    This determines a sheaf \( \mathcal F \) on \( X \) such that \( \mathcal F(B) = F(B) \) for all \( B \in \mathcal B \), agreeing with restriction maps.
+    Moreover, \( \mathcal F \) is unique up to unique isomorphism.
+\end{proposition}
+\begin{proof}
+    We first define the stalks using \( F \):
+    \[ \mathcal F_p = \faktor{\qty{(s_B, B) \mid p \in B \in \mathcal B, s_B \in F(B)}}{\sim} \]
+    We then use a sheafification idea to define \( \mathcal F(U) \).
+    The elements are the dependent functions \( f \in \prod_{p \in U} \mathcal F_p \) such that for each \( p \in U \), there exists a basic open set \( B \) containing \( p \) and a section \( s \in F(B) \) such that \( s_q = f_q \) in \( \mathcal F_q \) for all \( q \in B \).
+    This is then clearly a sheaf.
+    The natural maps \( F(B) \to \mathcal F(B) \) are isomorphisms by the sheaf axioms.
+\end{proof}

iii/alggeom/main.tex

@@ -16,5 +16,7 @@ \section{Introduction}
 \input{01_introduction.tex}
 \section{Sheaves}
 \input{02_sheaves.tex}
+\section{Schemes}
+\input{03_schemes.tex}
 
 \end{document}

iii/cat/03_adjunctions.tex

@@ -28,5 +28,135 @@ \subsection{Definition and examples}
         Applying this bijection is sometimes called \emph{currying} or \emph{\( \lambda \)-conversion}.
         We say that a category \( \mathcal C \) with binary products is \emph{cartesian closed} if \( (-) \times A : \mathcal C \to \mathcal C \) has a right adjoint, written \( [A, -] \) or \( (-)^A \), for each \( A \).
         For example, \( \mathbf{Cat} \) is cartesian closed, where \( \mathcal D^{\mathcal C} = [\mathcal C, \mathcal D] \) is the functor category that this notation already refers to.
+        \item An equivalence \( F : \mathcal C \to \mathcal D \), \( G : \mathcal D \to \mathcal C \) forms adjunctions both ways: \( F \dashv G, G \dashv F \).
+        \item Let \( \mathbf{Idem} \) be the category of pairs \( (A, e) \) where \( A \) is a set and \( e \) is an idempotent endomorphism \( A \to A \).
+        The morphisms in \( \mathbf{Idem} \) are the maps of sets which commute with the idempotents.
+        We have a functor \( F : \mathbf{Set} \to \mathbf{Idem} \) sending \( A \) to \( (A, 1_A) \).
+        Consider \( G : \mathbf{Idem} \to \mathbf{Set} \) sending \( (A, e) \) to the set of fixed points of \( e \).
+        Then \( F \dashv G \) since any morphism \( FA \to (B, e) \) takes values in \( G(B, e) \).
+        But also \( G \dashv F \), since a morphism \( (A, e) \to FB \) is entirely determined by its action on the fixed points in \( A \) under \( e \), because \( f(a) = f(ea) \).
+        This is not an equivalence of categories, because \( G \) is not faithful.
+        So not all pairs of functors that are adjoint in both directions form an equivalence.
+        \item Let \( \mathcal C \) be a category.
+        There is a unique functor \( G : \mathcal C \to \mathbf 1 \), where \( \mathbf 1 \) is the discrete category on a single object.
+        A left adjoint for \( G \), if it exists, sends the object in \( \mathbf 1 \) to an \emph{initial object} \( I \) of \( \mathcal C \), which is an object with a unique morphism to every object in \( \mathcal C \).
+        Dually, a right adjoint sends the object in \( \mathbf 1 \) to a \emph{terminal object} \( T \), which is an object with a unique morphism from every object in \( \mathcal C \).
+        In \( \mathbf{Set} \), the empty set is initial, and any singleton is terminal.
+        In \( \mathbf{Gp} \), the trivial group is initial and terminal.
+        \item Let \( f : A \to B \) be a function of sets, and let \( A' \subseteq A, B' \subseteq B \).
+        Then \( Pf(A') \subseteq B' \) if and only if \( A' \subseteq P^\star f(B') \).
+        Thus \( Pf \dashv P^\star f \) as functors between \( PA \) and \( PB \) as posets.
+        \item Let \( A, B \) be sets with a relation \( R \subseteq A \times B \).
+        We define mappings \( (-)^r : PA \to PB \) by
+        \[ S^r = \qty{b \in B \mid \forall a \in S,\, (a, b) \in R} \]
+        and \( (-)^\ell : PB \to PA \) by
+        \[ T^\ell = \qty{a \in A \mid \forall b \in T,\, (a, b) \in R} \]
+        These are contravariant functors, and
+        \[ S \subseteq T^\ell \iff S \times T \subseteq R \iff T \subseteq S^r \]
+        We say that \( (-)^\ell \) and \( (-)^r \) are \emph{adjoint on the right}.
+        This pair is called a \emph{Galois connection}.
+        \item The contravariant power-set functor \( P^\star \) is self-adjoint on the right, since functions \( A \to P^\star B \) and \( B \to P^\star A \) naturally correspond bijectively to subsets of \( A \times B \).
+        \item The dual vector space functor \( (-)^{\star\star} : \mathbf{Vect}_k \to \mathbf{Vect}_k \) is self-adjoint on the right, as linear maps \( V \to W^\star \) and linear maps \( W \to V^\star \) both naturally correspond to bilinear forms on \( V \times W \).
     \end{enumerate}
 \end{example}
+
+\subsection{???}
+\begin{theorem}
+    Let \( G : \mathcal D \to \mathcal C \) be a functor and \( A \in \ob \mathcal C \).
+    Write \( (A \downarrow G) \) for the category whose objects are pairs \( (B, f) \) where \( B \in \ob \mathcal D \) and \( f : A \to GB \) in \( \mathcal C \), and whose morphisms \( (B, f) \to (B', f') \) are morphisms \( g : B \to B' \) which commute with \( f, f' \):
+    % https://q.uiver.app/#q=WzAsMyxbMCwwLCJBIl0sWzEsMCwiR0IiXSxbMSwxLCJHQiciXSxbMCwxLCJmIl0sWzEsMiwiR2ciXSxbMCwyLCJmJyIsMl1d
+\[\begin{tikzcd}
+	A & GB \\
+	& {GB'}
+	\arrow["f", from=1-1, to=1-2]
+	\arrow["Gg", from=1-2, to=2-2]
+	\arrow["{f'}"', from=1-1, to=2-2]
+\end{tikzcd}\]
+    Then specifying a left adjoint for \( G \) is equivalent to specifying an initial object of \( (A \downarrow G) \) for each \( A \).
+\end{theorem}
+The category \( (A \downarrow G) \) is sometimes called a \emph{comma category}.
+\begin{proof}
+    Suppose \( F \dashv G \).
+    Then let \( \eta_A : A \to GFA \) correspond to the identity \( 1_{FA} \) under the adjunction.
+    We show that \( (FA, \eta_A) \) is initial in \( (A \downarrow G) \).
+    Indeed, given \( f : A \to GB \), then
+    % https://q.uiver.app/#q=WzAsMyxbMCwwLCJBIl0sWzEsMCwiR0ZBIl0sWzEsMSwiR0IiXSxbMCwxLCJcXGV0YV9BIl0sWzEsMiwiR2ciXSxbMCwyLCJmIiwyXV0=
+\[\begin{tikzcd}
+	A & GFA \\
+	& GB
+	\arrow["{\eta_A}", from=1-1, to=1-2]
+	\arrow["Gg", from=1-2, to=2-2]
+	\arrow["f"', from=1-1, to=2-2]
+\end{tikzcd}\]
+    commutes if and only if \( g \) is the morphism corresponding to \( f \) under the adjunction.
+    In particular, for any \( f \), there is a unique such \( g \).
+
+    Conversely, suppose \( (FA, \eta_A) \) is initial in \( (A \downarrow G) \) for each \( A \).
+    Then we define the action of \( F \) on objects by mapping \( A \) to \( FA \).
+    We make \( F \) into a functor by mapping \( f : A \to A' \) to the unique morphism that makes the following square commute.
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJBIl0sWzEsMCwiR0ZBIl0sWzEsMSwiR0ZBJyJdLFswLDEsIkEnIl0sWzAsMSwiXFxldGFfQSJdLFsxLDIsIkdGZiJdLFswLDMsImYiLDJdLFszLDIsIlxcZXRhX3tBJ30iLDJdXQ==
+\[\begin{tikzcd}
+	A & GFA \\
+	{A'} & {GFA'}
+	\arrow["{\eta_A}", from=1-1, to=1-2]
+	\arrow["GFf", from=1-2, to=2-2]
+	\arrow["f"', from=1-1, to=2-1]
+	\arrow["{\eta_{A'}}"', from=2-1, to=2-2]
+\end{tikzcd}\]
+    Functoriality of \( F \) follows from the uniqueness of \( Ff \).
+    The bijection between morphisms \( f : A \to GB \) and \( g : FA \to B \) sends \( f \) to the unique \( g \) giving \( (Gg)\eta_A = f \).
+    Naturality of the bijection in \( A \) was built in to the definition of \( F \) as a functor, and naturality in \( B \) is easy.
+\end{proof}
+\begin{corollary}
+    Let \( F, F' : \mathcal C \to \mathcal D \) be left adjoints to \( G : \mathcal D \to \mathcal C \).
+    Then \( F \simeq F' \) in \( [\mathcal C, \mathcal D] \).
+\end{corollary}
+\begin{proof}
+    \( (FA, \eta_A) \) and \( (F'A, \eta'_A) \) are both initial objects in \( (A \downarrow G) \), and so there is a unique isomorphism \( \alpha_A : (FA, \eta_A) \to (F'A, \eta'_A) \) in this category.
+    The map \( A \mapsto \alpha_A \) is natural, because given \( f : A \to A' \), \( \alpha_{A'}(Ff) \) and \( (F'f) \alpha_A \) are both morphisms \( (FA, \eta_A) \rightrightarrows (F'A', \eta'_{A'} f) \) from an initial object in \( (A \downarrow G) \), so must be equal.
+\end{proof}
+\begin{lemma}
+    Suppose
+    % https://q.uiver.app/#q=WzAsMyxbMCwwLCJcXG1hdGhjYWwgQyJdLFsxLDAsIlxcbWF0aGNhbCBEIl0sWzIsMCwiXFxtYXRoY2FsIEUiXSxbMCwxLCJGIiwwLHsib2Zmc2V0IjotMn1dLFsxLDIsIkgiLDAseyJvZmZzZXQiOi0yfV0sWzIsMSwiSyIsMCx7Im9mZnNldCI6LTJ9XSxbMSwwLCJHIiwwLHsib2Zmc2V0IjotMn1dXQ==
+\[\begin{tikzcd}
+	{\mathcal C} & {\mathcal D} & {\mathcal E}
+	\arrow["F", shift left=2, from=1-1, to=1-2]
+	\arrow["H", shift left=2, from=1-2, to=1-3]
+	\arrow["K", shift left=2, from=1-3, to=1-2]
+	\arrow["G", shift left=2, from=1-2, to=1-1]
+\end{tikzcd}\]
+    where \( F \dashv G \) and \( H \dashv K \).
+    Then \( HF \dashv GK \).
+\end{lemma}
+\begin{proof}
+    We have bijections between morphisms \( HFA \to C \), morphisms \( FA \to KC \), and morphisms \( A \to GKC \), which are natural in \( A \) and \( C \), so their composite is also natural.
+\end{proof}
+\begin{corollary}
+    Suppose the square of functors
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXG1hdGhjYWwgQyJdLFsxLDAsIlxcbWF0aGNhbCBEIl0sWzEsMSwiXFxtYXRoY2FsIEYiXSxbMCwxLCJcXG1hdGhjYWwgRSJdLFswLDEsIkYiXSxbMSwyLCJIIl0sWzAsMywiRyIsMl0sWzMsMiwiSyIsMl1d
+\[\begin{tikzcd}
+	{\mathcal C} & {\mathcal D} \\
+	{\mathcal E} & {\mathcal F}
+	\arrow["F", from=1-1, to=1-2]
+	\arrow["H", from=1-2, to=2-2]
+	\arrow["G"', from=1-1, to=2-1]
+	\arrow["K"', from=2-1, to=2-2]
+\end{tikzcd}\]
+    commutes, and all of the functors \( F, G, H, K \) have left adjoints \( F', G', H', K' \).
+    Then the square of left adjoints
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXG1hdGhjYWwgQyJdLFsxLDAsIlxcbWF0aGNhbCBEIl0sWzEsMSwiXFxtYXRoY2FsIEYiXSxbMCwxLCJcXG1hdGhjYWwgRSJdLFsxLDAsIkYnIiwyXSxbMiwxLCJIJyIsMl0sWzMsMCwiRyciXSxbMiwzLCJLJyJdXQ==
+\[\begin{tikzcd}
+	{\mathcal C} & {\mathcal D} \\
+	{\mathcal E} & {\mathcal F}
+	\arrow["{F'}"', from=1-2, to=1-1]
+	\arrow["{H'}"', from=2-2, to=1-2]
+	\arrow["{G'}", from=2-1, to=1-1]
+	\arrow["{K'}", from=2-2, to=2-1]
+\end{tikzcd}\]
+    commutes up to natural isomorphism.
+\end{corollary}
+This result holds for any shape of diagram, not just a square.
+The hypothesis can be weakened to only require that the first diagram commutes up to natural isomorphism.
+\begin{proof}
+    The two composites \( F'H' \) and \( G'K' \) are left adjoints to \( HF = KG \), so must be naturally isomorphic.
+\end{proof}

iii/mtncl/01_substructures.tex

@@ -194,7 +194,7 @@ \subsection{Universal theories and the method of diagrams}
 The diagram of a group is a slight generalisation of its multiplication table.
 Note that a model of a diagram is the same as an extension, and a model of an elementary diagram is the same as an elementary extension.
 \begin{lemma}
-    Let \( \mathcal T \) be a consistent theory, and let \( \mathcal T_\forall \) for the theory of universal sentences proven by \( \mathcal T \).
+    Let \( \mathcal T \) be a consistent theory, and let \( \mathcal T_\forall \) be the theory of universal sentences proven by \( \mathcal T \).
     If \( \mathcal N \) is a model of \( \mathcal T_\forall \), then \( \mathcal T \cup \operatorname{Diag} {\mathcal N} \) is consistent.
 \end{lemma}
 \begin{proof}