Merge branch 'main' of https://github.com/zeramorphic/cambridge-maths…

…-notes

iii/alggeom/05_modules_over_the_structure_sheaf.tex

@@ -159,7 +159,22 @@ \subsection{Coherent sheaves on projective schemes}
     \[ \mathcal F(d) = \mathcal F \otimes_{\mathcal O_X} \mathcal O_X(d) \]
     is globally generated.
 \end{theorem}
-As a consequence, every \( \mathcal F \) as above is a quotient of a vector bundle.
 \begin{proof}
     By formal properties, it is equivalent to show the statement for \( i_\star \mathcal F \); that is, \( i_\star \mathcal F(d) \) is globally generated on \( \mathbb P^n_R \).
+    Write \( \mathbb P^n_R = \Proj[x_0, \dots, x_n] \), and cover \( \mathbb P^n_R \) by \( U_i = \Spec B_i \) where \( B_i = R\qty[\frac{x_0}{x_i}] \).
+    We know that \( \eval{\mathcal F}_{U_i} = M_i^{\mathrm{sh}} \), and \( M_i \) is a finitely generated \( B_i \)-module.
+    Let \( \qty{s_{ij}} \) be generators for \( M_i \).
+    We claim that the sections \( \qty{x_i^d s_{ij}}_j \) of \( \eval{\mathcal F(d)}_{U_i}(U_i) \) are restrictions of global sections \( t_{ij} \) of \( \mathcal F(d) \) for sufficiently large \( d \).
+    Such \( d \) can be chosen to be independent of \( i \) and \( j \).
+    Indeed, if \( s_{ij} \) is an element of \( M_i = \mathcal F(U_i) \) and \( x_i \in \mathcal O_X(1) = \mathcal O_{\mathbb P^n_r}(1) \), we can show that \( x_i^d s_{ij} \in \mathcal (F \otimes \mathcal O(d))(U_i) \) is a restriction of a global section.
+
+    Now, on \( U_i \), the \( s_{ij} \) generate \( M_i^{\mathrm{sh}} \), but we have a morphism of sheaves \( \mathcal F \to \mathcal F(d) \), mapping \( s \) to \( x_i^d s \coloneq s \otimes x_i^d \).
+    This map is globally defined, but on \( U_i \) this restricts to an isomorphism \( \eval{\mathcal F}_{U_i} \to \eval{\mathcal F(d)}_{U_i} \) as \( x_i \) is invertible on \( U_i \).
+    Since the \( \qty{s_{ij}} \) generate \( \eval{\mathcal F}_{U_i} \), the \( x_i^d s_j \) generate \( \eval{\mathcal F(d)}_{U_i} \).
+    Thus, the \( t_{ij} \) globally generate \( \mathcal F(d) \).
 \end{proof}
+\begin{corollary}
+    Let \( i : X \rightarrowtail \mathbb P^n_R \) be a closed immersion.
+    Let \( \mathcal F \) be a coherent sheaf on \( X \).
+    Then \( \mathcal F \) is a quotient of \( \mathcal O(-d)^{\oplus N} \) for some sufficiently large \( N \) and some \( d \in \mathbb Z \).
+\end{corollary}

iii/alggeom/06_divisors.tex

@@ -0,0 +1,50 @@
+\subsection{Height and dimension}
+Recall that for a prime ideal \( \mathfrak p \) in \( R \), its \emph{height} is the largest \( n \) such that there exists a chain of inclusions of prime ideals
+\[ \mathfrak p_0 \subsetneq \mathfrak p_1 \subsetneq \dots \subsetneq \mathfrak p_n = \mathfrak p \]
+For example, if \( R \) is an integral domain, a prime ideal is of height 1 if and only if no nonzero prime ideal is strictly contained within it.
+\begin{example}
+    \begin{enumerate}
+        \item In any integral domain, \( (0) \) has height 0.
+        \item In \( \mathbb C[x, y] \), the ideal \( (x) \) has height 1, and the ideal \( (x, y) \) has height 2.
+    \end{enumerate}
+\end{example}
+It can be shown that in a unique factorisation domain, every prime ideal of height 1 is principal.
+
+We will globalise the notion of height 1 prime ideals, giving \emph{Weil divisors}, and also the notion of principal ideals, giving \emph{Cartier divisors}.
+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 \( X \).
+This point is often denoted by \( \eta \) or \( \eta_X \).
+\begin{definition}
+    Let \( X \) be a scheme.
+    \begin{enumerate}
+        \item The \emph{dimension} of \( X \) is the length \( n \) of the longest chain of nonempty closed irreducible subsets
+        \[ Z_0 \subsetneq Z_1 \subsetneq \dots \subsetneq Z_n \]
+        \item Let \( Z \subseteq X \) be closed and irreducible.
+        The \emph{codimension} of \( X \) is the length \( n \) of the longest chain
+        \[ Z = Z_0 \subsetneq Z_1 \subsetneq \dots \subsetneq Z_n \]
+        \item If \( X \) is a \emph{Noetherian topological space}, so every decreasing sequence of closed subsets stabilises, then every closed \( Z \subseteq X \) has a decomposition into finitely many irreducible closed subsets.
+        \item Suppose \( X \) is Noetherian, integral, and separated.
+        We say that \( X \) is \emph{regular in codimension 1} if for every subspace \( Y \subseteq X \) that is closed, irreducible, and of codimension 1, if \( \eta_Y \) denotes the generic point of \( Y \), then \( \mathcal O_{X, \eta_Y} \) is a discrete valuation ring, or equivalently a local principal ideal domain.
+    \end{enumerate}
+\end{definition}
+
+\subsection{Weil divisors}
+\begin{definition}
+    Let \( X \) be Noetherian, integral, separated, and regular in codimension 1.
+    A \emph{prime divisor} on \( X \) is an integral closed subscheme of codimension 1.
+    A \emph{Weil divisor} on \( X \) is an element of the free abelian group \( \operatorname{Div}(X) \) generated by the prime divisors.
+\end{definition}
+We will write \( D \in \operatorname{Div}(X) \) as \( \sum_i n_{Y_i} [Y_i] \) where the \( Y_i \) are prime divisors.
+\begin{definition}
+    A Weil divisor \( \sum_i n_{Y_i} [Y_i] \) is \emph{effective} if all \( n_{Y_i} \) are nonnegative.
+\end{definition}
+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 \( f \in \mathcal O_{X, \eta_X} = k(X) \).
+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.

iii/alggeom/main.tex

@@ -22,5 +22,7 @@ \section{Morphisms}
 \input{04_morphisms.tex}
 \section{Modules over the structure sheaf}
 \input{05_modules_over_the_structure_sheaf.tex}
+\section{Divisors}
+\input{06_divisors.tex}
 
 \end{document}

iii/cat/06_monoidal_and_enriched_categories.tex

@@ -41,6 +41,7 @@ \subsection{Monoidal categories}
 	\arrow["{\rho_A \otimes 1_B}"', from=1-1, to=2-2]
 \end{tikzcd}\]
     commute, and \( \lambda_I = \rho_I : I \otimes I \to I \).
+    A monoidal category is \emph{strict} if \( \alpha, \lambda, \rho \) are identities.
 \end{definition}
 \( \alpha \) is called the \emph{associator}, and \( \lambda \) and \( \rho \) are the \emph{left} and \emph{right unitors}.
 
@@ -196,3 +197,100 @@ \subsection{The coherence theorem}
 The theorem shows that for any two words \( w, w' \) involving the same set of variables without repetition, there is a unique natural isomorphism between \( w \) and \( w' \) obtained from compositions of instances of \( \alpha, \lambda, \gamma \) and their inverses.
 Note that \( \rho \) is not necessary, as it can be produced from instances of \( \lambda \) and \( \gamma \).
 The examples of monoidal categories above are all symmetric, except for (iv) and (v).
+
+\subsection{Monoidal functors}
+\begin{definition}
+    Let \( (\mathcal C, \otimes, I), (\mathcal D, \oplus, J) \) be monoidal categories.
+    A \emph{(lax) monoidal functor} \( F : (\mathcal C, \otimes, I) \to (\mathcal D, \oplus, J) \) is a functor \( F: \mathcal C \to \mathcal D \) equipped with a natural transformation \( \varphi_{A, B} : FA \oplus FB \to F(A \otimes B) \) and a morphism \( \iota : J \to FI \), such that the following diagrams commute.
+    % https://q.uiver.app/#q=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
+\[\begin{tikzcd}
+	{(FA \oplus FB) \oplus FC} & {F(A \otimes B) \oplus FC} & {F((A \otimes B) \otimes C)} \\
+	{FA \oplus (FB \oplus FC)} & {FA \oplus F(B \otimes C)} & {F(A \otimes (B \otimes C))}
+	\arrow["{\alpha_{FA,FB, FC}}"', from=1-1, to=2-1]
+	\arrow["{\varphi_{A,B} \oplus 1_{FC}}", from=1-1, to=1-2]
+	\arrow["{\varphi_{A \otimes B, C}}", from=1-2, to=1-3]
+	\arrow["{F\alpha_{A,B,C}}", from=1-3, to=2-3]
+	\arrow["{1_{FA} \oplus \varphi_{B,C}}"', from=2-1, to=2-2]
+	\arrow["{\varphi_{A,B \otimes C}}"', from=2-2, to=2-3]
+\end{tikzcd}\]
+% https://q.uiver.app/#q=WzAsNCxbMCwwLCJKIFxcb3BsdXMgRkEiXSxbMSwwLCJGSSBcXG9wbHVzIEZBIl0sWzEsMSwiRihJIFxcb3RpbWVzIEEpIl0sWzAsMSwiRkEiXSxbMCwxLCJcXGlvdGEgXFxvcGx1cyAxX3tGQX0iXSxbMSwyLCJcXHZhcnBoaV97SSxBfSJdLFswLDMsIlxcbGFtYmRhX3tGQX0iLDJdLFsyLDMsIkZcXGxhbWJkYV9BIl1d
+\[\begin{tikzcd}
+	{J \oplus FA} & {FI \oplus FA} \\
+	FA & {F(I \otimes A)}
+	\arrow["{\iota \oplus 1_{FA}}", from=1-1, to=1-2]
+	\arrow["{\varphi_{I,A}}", from=1-2, to=2-2]
+	\arrow["{\lambda_{FA}}"', from=1-1, to=2-1]
+	\arrow["{F\lambda_A}", from=2-2, to=2-1]
+\end{tikzcd}\quad\quad\begin{tikzcd}
+	{FA \oplus J} & {FA \oplus FI} \\
+	FA & {F(A \otimes I)}
+	\arrow["{1_{FA} \oplus \iota}", from=1-1, to=1-2]
+	\arrow["{\varphi_{A, I}}", from=1-2, to=2-2]
+	\arrow["{\rho_{FA}}"', from=1-1, to=2-1]
+	\arrow["{F\rho_A}", from=2-2, to=2-1]
+\end{tikzcd}\]
+    We say \( F \) is \emph{strong monoidal} (respectively \emph{strict monoidal}) if \( \varphi \) and \( \iota \) are isomorphisms (respectively identities).
+    An \emph{oplax} monoidal functor is the same definition, but where the directions of the maps \( \varphi \) and \( \iota \) are reversed.
+\end{definition}
+Note that the same letters are used for the associators and unitors in both monoidal categories.
+\begin{example}
+    \begin{enumerate}
+        \item The forgetful functor \( U : (\mathbf{AbGp}, \otimes, \mathbb Z) \to (\mathbf{Set}, \times, 1) \) is lax monoidal.
+        We define \( \iota : 1 \to \mathbb Z \) to map the element of \( 1 \) to the generator \( 1 \in \mathbb Z \), and define \( \varphi : UA \times UB \to U(A \otimes B) \) by \( (a, b) \mapsto a \otimes b \).
+        One can easily verify that the required diagrams commute.
+        \item The free functor \( F : (\mathbf{Set}, \times, 1) \to (\mathbf{AbGp}, \otimes, \mathbb Z) \) is strong monoidal, because \( F1 \cong \mathbb Z \) and \( F(A \times B) \cong FA \otimes FB \).
+        \item Let \( R \) be a commutative ring.
+        Then the forgetful functor \( \mathbf{Mod}_R \to \mathbf{AbGp} \) is lax monoidal, where \( \iota : \mathbb Z \to R \) is the natural map, and \( \varphi : A \otimes_{\mathbb Z} B \to A \otimes_R B \) is the quotient map.
+        Its left adjoint, the free functor \( \mathbf{AbGp} \to \mathbf{Mod}_R \), is strong monoidal.
+        \item If \( \mathcal C \) and \( \mathcal D \) have the cartesian monoidal structure, then any functor \( F : \mathcal C \to \mathcal D \) is oplax monoidal.
+        \( \iota : F1 \to 1 \) is the unique morphism to the terminal object of \( \mathcal D \), and \( \varphi_{A,B} : F(A \times B) \to FA \times FB \) is given by \( (F\pi_1, F\pi_2) \).
+        \( F \) is strong monoidal if and only if it preserves finite products.
+        \item If \( X \) and \( Y \) are metric spaces, then \( 1_{X \times Y} \) is non-expansive as a map \( (X \times Y, d_1) \to (X \times Y, d_\infty) \), making the identity functor \( 1_{\mathbf{Met}} \) into a monoidal functor \( (\mathbf{Met}, \times_\infty, 1) \to (\mathbf{Met}, \times_1, 1) \).
+        Note that the \( d_\infty \) metric on \( X \times Y \) defines the categorical product.
+    \end{enumerate}
+\end{example}
+\begin{lemma}
+    Let \( \mathcal C \) and \( \mathcal D \) be monoidal categories.
+    Let \( F \dashv G \), where \( F : \mathcal C \to \mathcal D \) and \( G : \mathcal D \to \mathcal C \).
+    Then there is a bijection between lax monoidal structures on \( G \) and oplax monoidal structures on \( F \).
+\end{lemma}
+\begin{proof}[Proof sketch]
+    Suppose we have \( (\varphi, \iota) \) on \( G \).
+    Then the transpose of \( \iota : J \to GI \) is a morphism \( FJ \to I \), and we have a natural transformation
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJGKEEgXFxvdGltZXMgQikiXSxbMSwwLCJGKEdGQSBcXG90aW1lcyBHRkIpIl0sWzIsMCwiRkcoRkEgXFxvcGx1cyBGQikiXSxbMywwLCJGQSBcXG9wbHVzIEZCIl0sWzAsMSwiRihcXGV0YV9BIFxcdGltZXMgXFxldGFfQikiXSxbMSwyLCJGIFxcdmFycGhpX3tGQSwgRkJ9Il0sWzIsMywiXFxlcHNpbG9uX3tGQSBcXG9wbHVzIEZCfSJdXQ==
+\[\begin{tikzcd}[column sep=large]
+	{F(A \otimes B)} & {F(GFA \otimes GFB)} & {FG(FA \oplus FB)} & {FA \oplus FB}
+	\arrow["{F(\eta_A \times \eta_B)}", from=1-1, to=1-2]
+	\arrow["{F \varphi_{FA, FB}}", from=1-2, to=1-3]
+	\arrow["{\epsilon_{FA \oplus FB}}", from=1-3, to=1-4]
+\end{tikzcd}\]
+    One can check that each of the required diagrams commute, defining an oplax monoidal structure on \( F \).
+    By duality, an oplax monoidal structure on \( F \) yields a lax monoidal structure on \( G \), and it can be shown that these constructions are inverse to each other.
+\end{proof}
+
+\subsection{Closed monoidal categories}
+\begin{definition}
+    We say that a monoidal category \( (\mathcal C, \otimes, I) \) is (\emph{left/right/bi})-\emph{closed} if \( A \otimes(-), (-) \otimes A \), or both have right adjoints for all \( A \).
+    If \( \otimes \) is symmetric, we say in any of these cases that \( \mathcal C \) is \emph{closed}.
+\end{definition}
+Right adjoints for \( (-) \otimes A \) are denoted \( [A, -] \) if they exist.
+\begin{example}
+    \begin{enumerate}
+        \item A cartesian closed category is a monoidal category with \( \otimes = \times \), that is closed as a monoidal category.
+        In particular, \( \mathbf{Set} \) and \( \mathbf{Cat} \) are cartesian closed.
+        \item The metric \( d_1 \) on the set \( [X, Y] \) of non-expansive maps \( X \to Y \) yields a closed structure on \( (\mathbf{Met}, \times_1, 1) \).
+        \item \( \mathbf{AbGp} \) and \( \mathbf{Mod}_R \) for any commutative ring \( R \) are monoidal closed, where \( [A, B] \) is the set of homomorphisms \( A \to B \), turned into an abelian group or \( R \)-module by pointwise addition and scalar multiplication.
+        The homomorphisms \( C \to [A,B] \) correspond under \( \lambda \)-conversion to bilinear maps \( C \times A \to B \), and thus to homomorphisms \( C \otimes_R A \to B \).
+        \item The cartesian monoidal structure on the category of pointed sets \( \mathbf{Set}_\star \) is not closed, but the monoidal structure given by the \emph{smash product} \( (-) \wedge (-) \) is closed, where
+        \[ (A, a_0) \wedge (B, b_0) = \faktor{A \times B}{\sim} \]
+        and \( \sim \) identifies all elements where either coordinate is the basepoint.
+        Basepoint-preserving maps \( A \wedge B \to C \) correspond to basepoint-preserving maps from \( A \) to the set \( [B,C] \) of basepoint-preserving maps \( B \to C \).
+        \item Consider the set \( \operatorname{Rel}(A \times A) = P(A \times A) \) of relations on \( A \).
+        This is a poset under inclusion, and is a monoid under relational composition.
+        Composition is order-preserving in each variable, making \( \operatorname{Rel}(A \times A) \) into a strict monoidal category.
+        It is not symmetric, but biclosed.
+        For the right adjoint to \( (-) \circ R \), we define \( R \Rightarrow T \) to be
+        \[ (R \Rightarrow T) = \qty{(b, c) \in A \times A \mid \forall a \in A,\, (a, b) \in R \Rightarrow (a, c) \in T} \]
+        Then \( S \subseteq (R \Rightarrow T) \) if and only if \( S \circ R \subseteq T \).
+    \end{enumerate}
+\end{example}