Lectures 09B

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}

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}

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}

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}