Lectures 21B

iii/alggeom/06_divisors.tex

@@ -14,7 +14,8 @@ \subsection{Height and dimension}
 In the case of Weil divisors, we will assume that the ambient scheme \( X \) is Noetherian, integral, separated, and \emph{regular in codimension 1}.
 
 If \( X \) is integral and \( U = \Spec A \) is an open affine, then the ideal \( (0) \subseteq A \) is called the \emph{generic point} of \( X \).
-The generic points given by each \( U \) coincide in \( X \).
+Each open affine is dense as they are irreducible, so they have a nontrivial intersection, including their generic points.
+The generic points given by each \( U \) therefore coincide in \( X \).
 This point is often denoted by \( \eta \) or \( \eta_X \).
 \begin{definition}
     Let \( X \) be a scheme.
@@ -43,8 +44,78 @@ \subsection{Weil divisors}
 If \( X \) is integral, for \( \Spec A = U \subseteq X \), the local ring \( \mathcal O_{X, \eta} \) is a field, as it is in particular the fraction field of \( A \).
 Indeed, because \( \eta \) is contained in every open affine, \( \mathcal O_{X, \eta} \) permits arbitrary denominators.
 
-Let \( f \in \mathcal O_{X, \eta_X} = k(X) \).
+Let \( f \in \mathcal O_{X, \eta_X} = k(X) \) be nonzero.
 Since for every prime divisor \( Y \subseteq X \), the ring \( \mathcal O_{X, \eta_Y} \) is a discrete valuation ring, we can calculate the valuation \( \nu_Y(f) \) of \( f \) in this ring.
 We thus define the divisor
 \[ \operatorname{div}(f) = \sum_{Y \subseteq X \text{ prime}} \nu_Y(f) [Y] \]
 We claim that this is a Weil divisor; that is, the sum is finite.
+\begin{proposition}
+    The sum
+    \[ \sum_{Y \subseteq X \text{ prime}} \nu_Y(f) [Y] \]
+    is finite.
+\end{proposition}
+\begin{proof}
+    Let \( f \in k(X)^\times \), and choose \( A \) such that \( U = \Spec A \) is an affine open, so \( FF(A) = k(X) \). 
+    We can also require that \( f \in A \) by localising at the denominator, so \( f \) is \emph{regular} on \( U \).
+    Then \( X \setminus U \) is closed and of codimension at least 1, so only finitely many prime Weil divisors \( Y \) of \( X \) are contained in \( X \setminus U \).
+    On \( U \), as \( f \) is regular, \( \nu_Y(f) \geq 0 \) for all \( Y \).
+    But \( \nu_Y(f) > 0 \) if and only if \( Y \) is contained in \( \mathbb V(f) \subseteq U \), and by the same argument, there are only finitely many such \( Y \).
+\end{proof}
+\begin{definition}
+    A Weil divisor of the form \( \operatorname{div}(f) \) is called \emph{principal}.
+\end{definition}
+In \( \operatorname{Div}(X) \), the set of principal divisors form a subgroup \( \operatorname{Prin}(X) \), and we define the \emph{Weil divisor class group} of \( X \) to be
+\[ \operatorname{Cl}(X) = \faktor{\operatorname{Div}(X)}{\operatorname{Prin}(X)} \]
+\begin{remark}
+    \begin{enumerate}
+        \item Let \( A \) be a Noetherian domain.
+        Then \( A \) is a unique factorisation domain if and only if \( A \) is integrally closed and \( \operatorname{Cl}(\Spec A) \) is trivial.
+        This is related to the fact that in unique factorisation domains, all primes of height 1 are principal.
+        In particular, there exist rings with nontrivial class groups of their spectra.
+        \item \( \operatorname{Cl}(\mathbb A^n_k) = 0 \).
+        \item \( \operatorname{Cl}(\mathbb P^n_k) \cong \mathbb Z \); we will prove this shortly.
+        % TODO: break here, organise into props/proofs
+        \item Let \( Z \subseteq X \) is closed, and let \( U = X \setminus Z \).
+        Then there is a surjective map \( \operatorname{Cl}(X) \twoheadrightarrow \operatorname{Cl}(U) \), defined by \( [Y] \mapsto [Y \cap U] \), but instead mapping \( [Y] \) to zero if \( Y \cap U = \varnothing \).
+        This is well-defined, as \( k(X) \) and \( k(U) \) are naturally isomorphic, so principal divisors are mapped to principal divisors.
+        For surjectivity, note that given a prime Weil divisor \( D \subseteq U \), its closure \( \overline D \) in \( X \) is a prime Weil divisor that restricts to \( D \) under the map.
+        \item If \( Z \) has codimension at least 2, then \( \operatorname{Cl}(X) \twoheadrightarrow \operatorname{Cl}(U) \) is an isomorphism.
+        This is because \( Z \) does not enter the definition of \( \operatorname{Cl}(X) \).
+        \item If \( Z \subseteq X \) is integral, closed, and of codimension 1, there is an exact sequence
+        % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXG1hdGhiYiBaIl0sWzEsMCwiXFxvcGVyYXRvcm5hbWV7Q2x9KFgpIl0sWzIsMCwiXFxvcGVyYXRvcm5hbWV7Q2x9KFUpIl0sWzMsMCwiMCJdLFswLDEsIjEgXFxtYXBzdG8gW1pdIl0sWzEsMl0sWzIsM11d
+        \[\begin{tikzcd}
+            {\mathbb Z} & {\operatorname{Cl}(X)} & {\operatorname{Cl}(U)} & 0
+            \arrow["{1 \mapsto [Z]}", from=1-1, to=1-2]
+            \arrow[from=1-2, to=1-3]
+            \arrow[from=1-3, to=1-4]
+        \end{tikzcd}\]
+        called the \emph{excision} exact sequence.
+        Indeed, the kernel of \( \operatorname{Cl}(X) \to \operatorname{Cl}(U) \) are exactly the divisors in \( X \) contained in \( Z \).
+    \end{enumerate}
+\end{remark}
+\begin{proposition}
+    Let \( k \) be a field.
+    Then, \( \operatorname{Cl}(\mathbb P^n_k) \cong \mathbb Z \).
+\end{proposition}
+\begin{proof}
+    Let \( D \subseteq \mathbb P^n \) be integral, closed, and of codimension 1.
+    Then \( D = \mathbb V(f) \) where \( f \) is homogeneous of some degree \( d \); we will define \( \deg(D) = d \).
+    We extend linearly to obtain a homomorphism \( \deg : \operatorname{Div}(\mathbb P^n_k) \to \mathbb Z \).
+    We claim that this gives an isomorphism \( \operatorname{Cl}(\mathbb P^n_k) \to \mathbb Z \).
+    First, this is well defined on classes, since if \( f = \frac{g}{h} \) is a rational function, then \( g \) and \( h \) are homogeneous polynomials of the same degree, so \( \deg(\operatorname{div}(f)) = 0 \).
+    This is surjective, by taking \( H = \mathbb V(x_0) \) for \( x_0 \) homogeneous linear.
+    For injectivity, suppose \( D = \sum n_{Y_i} [Y_i] \) with \( \sum n_{Y_i} \deg(Y_i) = 0 \).
+    Write \( Y_i = \mathbb V(g_i) \), and let \( f = \prod g_i^{n_{Y_i}} \).
+    Now \( f \) is a homogeneous rational function of degree zero.
+\end{proof}
+
+\subsection{Cartier divisors}
+Let \( X \) be a scheme.
+Consider the presheaf on \( X \) given by mapping \( U = \Spec A \) to \( S^{-1}A \) where \( S \) is the set of all elements that are not zero divisors.
+Sheafification yields the sheaf of rings \( \mathcal K_X \).
+Define \( \mathcal K_X^\star \subseteq \mathcal K_X \) to be the subsheaf of invertible elements; this is a sheaf of abelian groups under multiplication.
+Similarly, let \( \mathcal O_X^\star \subseteq \mathcal O_X \) be the subsheaf of invertible elements.
+Thus, every section of \( \faktor{\mathcal K_X^\star}{\mathcal O_X^\star} \) can be prescribed by \( \qty{(U_i, f_i)} \) where \( U_i \) is a cover of \( X \), \( f_i \) is a section of \( \mathcal K_X^\star(U_i) \), and that on \( U_i \cap U_j \), the ratio \( \faktor{f_i}{f_j} \) lies in \( \mathcal O_X^\star(U_i \cap U_j) \).
+\begin{definition}
+    A \emph{Cartier divisor} is a section of the sheaf \( \faktor{\mathcal K_X^\star}{\mathcal O_X^\star} \).
+\end{definition}

iii/cat/06_monoidal_and_enriched_categories.tex

@@ -294,3 +294,104 @@ \subsection{Closed monoidal categories}
         Then \( S \subseteq (R \Rightarrow T) \) if and only if \( S \circ R \subseteq T \).
     \end{enumerate}
 \end{example}
+
+\subsection{Enriched categories}
+\begin{definition}
+    Let \( (\mathcal E, \otimes, I) \) be a monoidal category.
+    An \emph{\( \mathcal E \)-enriched category} is
+    \begin{enumerate}
+        \item a collection \( \ob \mathcal C \) of objects;
+        \item an object \( \mathcal C(A, B) \) of \( \mathcal E \) for each pair of objects \( A, B \in \ob \mathcal C \);
+        \item morphisms \( \iota_A : I \to \mathcal C(A, A) \) for each \( A \);
+        \item morphisms \( \kappa_{A,B,C} : \mathcal C(B, C) \otimes \mathcal C(A, B) \to \mathcal C(A, C) \) for objects \( A, B, C \),
+    \end{enumerate}
+    such that the following diagrams commute.
+    % https://q.uiver.app/#q=WzAsMyxbMCwwLCJJIFxcb3RpbWVzIFxcbWF0aGNhbCBDKEEsIEIpIl0sWzEsMCwiXFxtYXRoY2FsIEMoQiwgQikgXFxvdGltZXMgXFxtYXRoY2FsIEMoQSwgQikiXSxbMSwxLCJcXG1hdGhjYWwgQyhBLCBCKSJdLFswLDEsIlxcaW90YV9CIFxcb3RpbWVzIDFfe1xcbWF0aGNhbCBDKEEsIEIpfSJdLFsxLDIsIlxca2FwcGFfe0EsQixCfSJdLFswLDIsIlxcbGFtYmRhX3tcXG1hdGhjYWwgQyhBLCBCKX0iLDJdXQ==
+\[\begin{tikzcd}
+	{I \otimes \mathcal C(A, B)} & {\mathcal C(B, B) \otimes \mathcal C(A, B)} \\
+	& {\mathcal C(A, B)}
+	\arrow["{\iota_B \otimes 1_{\mathcal C(A, B)}}", from=1-1, to=1-2]
+	\arrow["{\kappa_{A,B,B}}", from=1-2, to=2-2]
+	\arrow["{\lambda_{\mathcal C(A, B)}}"', from=1-1, to=2-2]
+\end{tikzcd}\]
+% https://q.uiver.app/#q=WzAsMyxbMCwwLCJcXG1hdGhjYWwgQyhBLCBCKSBcXG90aW1lcyBJIl0sWzEsMCwiXFxtYXRoY2FsIEMoQSwgQikgXFxvdGltZXMgXFxtYXRoY2FsIEMoQSwgQSkiXSxbMSwxLCJcXG1hdGhjYWwgQyhBLCBCKSJdLFswLDEsIlxcaW90YV9CIFxcb3RpbWVzIDFfe1xcbWF0aGNhbCBDKEEsIEIpfSJdLFsxLDIsIlxca2FwcGFfe0EsQSxCfSJdLFswLDIsIlxccmhvX3tcXG1hdGhjYWwgQyhBLCBCKX0iLDJdXQ==
+\[\begin{tikzcd}
+	{\mathcal C(A, B) \otimes I} & {\mathcal C(A, B) \otimes \mathcal C(A, A)} \\
+	& {\mathcal C(A, B)}
+	\arrow["{\iota_B \otimes 1_{\mathcal C(A, B)}}", from=1-1, to=1-2]
+	\arrow["{\kappa_{A,A,B}}", from=1-2, to=2-2]
+	\arrow["{\rho_{\mathcal C(A, B)}}"', from=1-1, to=2-2]
+\end{tikzcd}\]
+% https://q.uiver.app/#q=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
+\[\begin{tikzcd}
+	{(\mathcal C(C, D) \otimes \mathcal C(B, C)) \otimes \mathcal C(A, B)} & {\mathcal C(B, D) \otimes \mathcal C(A, B)} \\
+	&& {\mathcal C(A, D)} \\
+	{\mathcal C(C, D) \otimes (\mathcal C(B, C) \otimes \mathcal C(A, B))} & {\mathcal C(C, D) \otimes \mathcal C(A, C)}
+	\arrow["{\kappa \otimes 1}", from=1-1, to=1-2]
+	\arrow["\kappa", from=1-2, to=2-3]
+	\arrow["\alpha"', from=1-1, to=3-1]
+	\arrow["{1 \otimes \kappa}"', from=3-1, to=3-2]
+	\arrow["\kappa"', from=3-2, to=2-3]
+\end{tikzcd}\]
+\end{definition}
+\begin{definition}
+    Let \( \mathcal C, \mathcal D \) be \( \mathcal E \)-enriched categories.
+    An \emph{\( \mathcal E \)-enriched functor} \( \mathcal C \to \mathcal D \) consists of a map of objects \( F : \ob \mathcal C \to \ob \mathcal D \) together with morphisms \( F_{A,B} : \mathcal C(A, B) \to \mathcal D(FA, FB) \) for each pair of objects \( A, B \in \ob \mathcal C \), in such a way that is compatible with identities and composition.
+\end{definition}
+\begin{definition}
+    Let \( F, G : \mathcal C \rightrightarrows \mathcal D \) be \( \mathcal E \)-enriched functors between \( \mathcal E \)-enriched categories.
+    An \emph{\( \mathcal E \)-enriched natural transformation} \( F \to G \) assigns a morphism \( \theta_A : I \to \mathcal D(FA, GA) \) to each \( A \in \ob \mathcal C \), satisfying the naturality condition
+    % https://q.uiver.app/#q=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
+\[\begin{tikzcd}
+	{\mathcal C(A, B)} & {\mathcal D(FA, FB)} & {I \otimes D(FA, FB)} \\
+	{\mathcal D(GA, GB)} && {\mathcal D(FB, GB) \otimes \mathcal D(FA, FB)} \\
+	{\mathcal D(GA, GB) \otimes I} & {\mathcal D(GA, GB) \otimes \mathcal D(FA, GA)} & {\mathcal D(FA, GB)}
+	\arrow["{F_{A,B}}", from=1-1, to=1-2]
+	\arrow["{\lambda^{-1}}", from=1-2, to=1-3]
+	\arrow["{\theta_B \otimes 1}", from=1-3, to=2-3]
+	\arrow["\kappa", from=2-3, to=3-3]
+	\arrow["{G_{A,B}}"', from=1-1, to=2-1]
+	\arrow["{\rho^{-1}}"', from=2-1, to=3-1]
+	\arrow["{1 \otimes \theta_A}"', from=3-1, to=3-2]
+	\arrow["\kappa"', from=3-2, to=3-3]
+\end{tikzcd}\]
+\end{definition}
+If \( \mathcal C \) is an \( \mathcal E \)-enriched category, its \emph{underlying ordinary category} \( \abs{\mathcal C} \) is the category where the objects are those of \( \mathcal C \), the morphisms \( A \to B \) are the morphisms \( I \to \mathcal C(A, B) \) in \( \mathcal E \), where the identity morphisms are given by \( \iota_A \), and the composition of \( g : C \to B \) and \( f : A \to B \) given by
+\[\begin{tikzcd}
+	I & {I \otimes I} & {\mathcal C(B, C) \otimes \mathcal C(A, B)} & {\mathcal C(A, C)}
+	\arrow["{\lambda_I^{-1}}", from=1-1, to=1-2]
+	\arrow["{g \otimes f}", from=1-2, to=1-3]
+	\arrow["\kappa", from=1-3, to=1-4]
+\end{tikzcd}\]
+One can check that this indeed forms a category.
+An \emph{\( \mathcal E \)-enrichment} of an ordinary category \( \mathcal C_0 \) is an \( \mathcal E \)-enriched category \( \mathcal C \) such that \( \abs{\mathcal C} \cong \mathcal C_0 \).
+\begin{example}
+    \begin{enumerate}
+        \item A category enriched over \( (\mathbf{Set}, \times, 1) \) is a locally small category.
+        \item A category enriched over the poset \( 2 = \qty{0,1} \) with \( 0 < 1 \) is a preorder.
+        \item A category enriched over \( (\mathbf{Cat}, \times, \mathbf{1}) \) is a \emph{2-category}.
+        Its morphisms or \emph{1-arrows} \( A \to B \) are the objects of a category \( \mathcal C(A, B) \).
+        It has \emph{2-arrows} between parallel pairs \( f, g : A \rightrightarrows B \), which are the morphisms \( f \to g \) in the category \( \mathcal C(A, B) \).
+        \( \mathbf{Cat} \) is a 2-category, by taking the 2-arrows to be the natural transformations.
+        The category of small \( \mathcal E \)-enriched categories with \( \mathcal E \)-enriched functors is a 2-category.
+        \item A category enriched over \( (\mathbf{AbGp}, \otimes, \mathbb Z) \) is an \emph{additive category}.
+        \item If \( \mathcal E \) is a right closed monoidal category, it has a canonical enrichment structure over itself.
+        Take \( \mathcal E(A, B) \) to be \( [A, B] \), where \( [A, -] \) is the right adjoint of \( (-) \otimes A \).
+        The identity \( I \to [A, A] \) is the transpose \( \lambda_A : I \otimes A \to A \), and the composition \( \kappa \) is the transpose of
+        % https://q.uiver.app/#q=WzAsNCxbMCwwLCIoW0IsIENdIFxcb3RpbWVzIFtBLCBCXSkgXFxvdGltZXMgQSJdLFsxLDAsIltCLENdIFxcb3RpbWVzIChbQSxCXSBcXG90aW1lcyBBKSJdLFsyLDAsIltCLENdIFxcb3RpbWVzIEIiXSxbMywwLCJDIl0sWzAsMSwiXFxhbHBoYSJdLFsxLDIsIjEgXFxvdGltZXMgXFxtYXRocm17ZXZ9Il0sWzIsMywiXFxtYXRocm17ZXZ9Il1d
+\[\begin{tikzcd}
+	{([B, C] \otimes [A, B]) \otimes A} & {[B,C] \otimes ([A,B] \otimes A)} & {[B,C] \otimes B} & C
+	\arrow["\alpha", from=1-1, to=1-2]
+	\arrow["{1 \otimes \mathrm{ev}}", from=1-2, to=1-3]
+	\arrow["{\mathrm{ev}}", from=1-3, to=1-4]
+\end{tikzcd}\]
+        where \( \mathrm{ev} \) is the evaluation map, which is precisely the counit of the adjunction.
+        \item A one-object \( \mathcal E \)-enriched category is an \emph{(internal) monoid} in \( \mathcal E \); it consists of an object \( M \) of \( \mathcal E \), equipped with morphisms \( e : I \to M \) and \( m : M \otimes M \to M \) satisfying the left and right unit laws and the associativity law.
+        \begin{enumerate}
+            \item An internal monoid in \( \mathbf{Set} \) is a monoid.
+            \item An internal monoid in \( \mathbf{AbGp} \) is a ring.
+            \item An internal monoid in \( \mathbf{Cat} \) is a strict monoidal category.
+            \item An internal monoid in \( [\mathcal C, \mathcal C] \) is a monad on \( \mathcal C \).
+        \end{enumerate}
+    \end{enumerate}
+\end{example}

iii/cat/07_additive_and_abelian_categories.tex

@@ -0,0 +1,25 @@
+\subsection{Additive categories}
+In this section, we will study categories enriched over \( (\mathbf{AbGp}, \otimes, \mathbb Z) \); these are called \emph{additive} categories.
+We will also consider other weaker enrichments: a category enriched over \( (\mathbf{Set}_\star, \wedge, 2) \) is called \emph{pointed}, and a category enriched over \( (\mathbf{CMon}, \otimes, \mathbb N) \), where \( \mathbf{CMon} \) is the category of commutative monoids, is called \emph{semi-additive}.
+
+In a pointed category \( \mathcal C \), each \( \mathcal C(A, B) \) has a distinguished element 0, and all composites with zero morphisms are zero morphisms.
+In a semi-additive category \( \mathcal C \), each \( \mathcal C(A, B) \) has a binary addition operation which is associative, commutative, and has an identity \( 0 \).
+Composition in a semi-additive category is bilinear, so \( (f + g)(h + k) = fh + gh + fk + gk \) whenever the composites are defined.
+In an additive category, each morphism \( f \in \mathcal C(A, B) \) has an additive inverse \( -f \in \mathcal C(A, B) \).
+\begin{lemma}
+    \begin{enumerate}
+        \item For an object \( A \) in a pointed category \( \mathcal C \), the following are equivalent.
+        \begin{enumerate}
+            \item \( A \) is a terminal object of \( \mathcal C \).
+            \item \( A \) is an initial object of \( \mathcal C \).
+            \item \( 1_A = 0 : A \to A \).
+        \end{enumerate}
+        \item For objects \( A, B, C \) in a semi-additive category \( \mathcal C \), the following are equivalent.
+        \begin{enumerate}
+            \item there exist morphisms \( \pi_1 : C \to A \) and \( \pi_2 : C \to B \) making \( C \) into a product of \( A \) and \( B \);
+            \item there exist morphisms \( \nu_1 : A \to C \) and \( \nu_2 : B \to C \) making \( C \) into a coproduct of \( A \) and \( B \);
+            \item there exist morphisms \( \pi_1 : C \to A, \pi_2 : C \to B, \nu_1 : A \to C, \nu_2 : B \to C \) satisfying
+            \[ \pi_1 \nu_1 = 1_A;\quad \pi_2 \nu_2 = 1_B;\quad \pi_1 \nu_2 = 0;\quad \pi_2 \nu_1 = 0;\quad \nu_1 \pi_1 + \nu_2 \pi_1 = 1_C \]
+        \end{enumerate}
+    \end{enumerate}
+\end{lemma}

iii/cat/main.tex

@@ -22,5 +22,7 @@ \section{Monads}
 \input{05_monads.tex}
 \section{Monoidal and enriched categories}
 \input{06_monoidal_and_enriched_categories.tex}
+\section{Additive and abelian categories}
+\input{07_additive_and_abelian_categories.tex}
 
 \end{document}