Lectures 11B

iii/alggeom/03_schemes.tex

@@ -244,8 +244,6 @@ \subsection{Gluing schemes}
 % exercise: \mathcal O_{\mathbb P^n_A}(\mathbb P^n_A) = A.
 
 \subsection{The Proj construction}
-% Idea:
-% Spec : Rings -> Schemes; Proj : GradedRings -> Schemes
 \begin{definition}
     A \emph{\( \mathbb Z \)-grading} on a ring \( A \) is a decomposition
     \[ A = \bigoplus_{i \in \mathbb Z} A_i \]
@@ -260,4 +258,65 @@ \subsection{The Proj construction}
     % how?
 \end{example}
 Note that by definition, \( A_0 \) is a subring of \( A \).
-We will always assume that the degree 1 elements of a graded ring generate \( A \) as an algebra over \( A_0 \).
+For simplicity, we will always assume in this course that the degree 1 elements of a graded ring generate \( A \) as an algebra over \( A_0 \).
+We also typically assume that \( A_i = 0 \) for \( i < 0 \).
+We define
+\[ A_+ = \bigoplus_{i \geq 1} A_i \subseteq A \]
+This forms an ideal in \( A \), called the \emph{irrelevant ideal}.
+If \( A \) is a polynomial ring with the usual grading, the irrelevant ideal corresponds to the point \( \vb 0 \) in the theory of varieties.
+This aligns with the definition of projective space in classical algebraic geometry, in which the point \( \vb 0 \) is deleted.
+
+A \emph{homogeneous element} \( f \in A \) is an element contained in some \( A_d \).
+An ideal \( I \) of \( A \) is called \emph{homogeneous} if it is generated by homogeneous elements.
+\begin{definition}
+    Let \( A \) be a graded ring.
+    \( \Proj A \) is the set of homogeneous prime ideals in \( A \) that do not contain the irrelevant ideal.
+    If \( I \subseteq A \) is homogeneous, we define
+    \[ \mathbb V(I) = \qty{\mathfrak p \in \Proj A \mid I \subseteq \mathfrak p} \]
+    The \emph{Zariski topology} on \( \Proj A \) is the topology where the closed sets are of the form \( \mathbb V(I) \) where \( I \) is a homogeneous ideal.
+\end{definition}
+The Spec construction allows us to convert rings into schemes; the Proj construction allows us to convert graded rings into schemes.
+Unlike Spec, the construction of Proj is not functorial.
+
+Let \( f \in A_1 \) and \( U_f = \Proj A \setminus \mathbb V(f) \).
+Observe that the set \( \qty{U_f}_{f \in A_1} \) covers \( \Proj A \), because the \( f \) generate the unit ideal.
+The ring \( A\qty[\frac{1}{f}] = A_f \) is naturally \( \mathbb Z \)-graded by defining \( \deg \frac{1}{f} = -\deg f \).
+Note that \( A_f \) may have negatively graded elements, even though \( A \) does not.
+\begin{example}
+    Let \( A = k[x_0, x_1] \) and \( f = x_0 \).
+    Then in \( A\qty[\frac{1}{f}] = k[x_0, x_1, x_0^{-1}] \), the degree zero elements include \( k \) and elements such as \( \frac{x_1}{x_0}, \frac{x_1^2 + x_1 x_0}{x_0^2} \).
+    There are degree one elements such as \( \frac{x_1^2}{x_0} \).
+\end{example}
+\begin{proposition}
+    There is a natural bijection
+    \[ \qty{\text{homogeneous prime ideals in \( A \) that miss \( f \)}} \leftrightarrow \qty{\text{prime ideals in } (A_f)_0} \]
+\end{proposition}
+Note also that the set of homogeneous prime ideals in \( A \) that miss \( f \) are naturally in bijection with the homogeneous prime ideals in \( A_f \).
+\begin{proof}
+    Suppose \( \mathfrak q \) is a prime ideal in \( \qty(A\qty[\frac{1}{f}])_0 \).
+    Then let \( \psi(\mathfrak q) \) be the ideal
+    \[ \psi(\mathfrak q) = \qty( \bigcup_{d \geq 0} \qty{a \in A_d \midd \frac{a}{f^d} \in \mathfrak q} \subseteq A ) \]
+    One can check that this is prime.
+    Now suppose \( \mathfrak p \) is a homogeneous prime ideal missing \( f \).
+    Define \( \varphi(\mathfrak p) \) to be
+    \[ \varphi(\mathfrak p) = \qty(p \cdot A\qty[\frac{1}{f}] \cap \qty(A\qty[\frac{1}{f}])_0) \]
+    This ideal is also prime.
+
+    One can easily check that \( \varphi \circ \psi \) is the identity.
+    For the other direction, suppose \( \mathfrak p \) is a homogeneous prime ideal missing \( f \); we show that \( \mathfrak p = \psi(\varphi(\mathfrak p)) \) by antisymmetry.
+    If \( a \in \mathfrak p \in A_d \), then \( \frac{a}{f^d} \in \varphi(\mathfrak p) \), so \( a \in \psi(\varphi(\mathfrak p)) \) by construction.
+    Conversely, if \( a \in \psi(\varphi(\mathfrak p)) \), then \( \frac{a}{f^d} \in \varphi(\mathfrak p) \) for some \( d \), so there exists \( b \in \mathfrak p \) such that \( \frac{b}{f^e} = \frac{a}{f^d} \) in \( A\qty[\frac{1}{f}] \).
+    Hence for some \( k \geq 0 \), we have \( f^k (f^d b - f^e a) = 0 \), and \( f^{e+k} \notin \mathfrak p \).
+    But by primality, \( a \in \mathfrak p \), as required.
+\end{proof}
+The bijection constructed is compatible with ideal containment, so is a homeomorphism of topological spaces
+\[ U_f \leftrightarrow \Spec (A_f)_0 \]
+Thus \( \Proj A \) is covered by open sets homeomorphic to an affine scheme.
+If \( f, g \in A_1 \), then \( U_f \cap U_g \) is naturally homeomorphic to
+\[ \qty(\Spec A\qty[\frac{1}{f}])_0\qty[\frac{f}{g}] = \Spec \qty(A\qty[f^{-1}, g^{-1}])_0 \]
+Take the open cover \( \qty{U_f} \) with structure sheaf \( \mathcal O_{\Spec (A_f)_0} \) on each \( U_f \), and isomorphisms on \( U_f \cap U_g \) by the condition above.
+The cocycle condition follows from the formal properties of the localisation.
+Therefore, \( \Proj A \) is a scheme.
+
+If \( A = k[x_0, \dots, x_n] \) with the standard grading, we write \( \mathbb P^n_k \) for \( \Proj A \).
+

iii/alggeom/04_morphisms.tex

@@ -0,0 +1,8 @@
+\subsection{???}
+Let \( (X, \mathcal O_X) \) be a scheme.
+The stalks \( \mathcal O_{X,\mathfrak p} \) are local rings: they have a unique maximal ideal, which is the set of all non-unit elements.
+Given \( f \in \mathcal O_X(U) \), we can meaningfully ask whether \( f \) vanishes at \( \mathfrak p \); that is, if the image of \( f \) in \( \mathcal O_{X, \mathfrak p} \) is contained in the maximal ideal.
+\begin{definition}
+    A morphism of ringed spaces \( f : (X, \mathcal O_X) \to (Y, \mathcal O_Y) \) consists of a continuous function \( f : X \to Y \) and a morphism \( f^\sharp : \mathcal O_Y \to f_\star \mathcal O_X \) between sheaves of rings on \( Y \).
+\end{definition}
+\( f^\sharp \) represents function composition with \( f^{-1} \), although the ring \( \mathcal O_X \) may not be a ring of functions.

iii/alggeom/main.tex

@@ -18,5 +18,7 @@ \section{Sheaves}
 \input{02_sheaves.tex}
 \section{Schemes}
 \input{03_schemes.tex}
+\section{Morphisms}
+\input{04_morphisms.tex}
 
 \end{document}

iii/cat/04_limits.tex

@@ -281,3 +281,108 @@ \subsection{Preservation and creation}
         A free product of two groups \( G, H \) is always nonabelian, except for the case where either \( G \) or \( H \) is the trivial group, but the coproduct of the trivial group with \( H \) is isomorphic to \( H \) in both categories.
     \end{enumerate}
 \end{example}
+\begin{lemma}
+    Suppose \( \mathcal D \) has limits of shape \( J \).
+    Then, for any \( \mathcal C \), the functor category \( [\mathcal C, \mathcal D] \) also has limits of shape \( J \), and the forgetful functor \( [\mathcal C, \mathcal D] \to \mathcal D^{\ob \mathcal C} \) creates them.
+\end{lemma}
+\begin{proof}
+    Given a diagram \( D : J \to [\mathcal C, \mathcal D] \), we can regard it as a functor \( D : J \times \mathcal C \to \mathcal D \), so for a fixed object in \( \mathcal C \), we obtain a diagram \( D(-, A) \) of shape \( J \) in \( \mathcal D \), which has a limit \( (LA, (\lambda_{j,A})_{j \in \ob J}) \).
+    Given any \( f : A \to B \) in \( \mathcal C \), the composites
+    % https://q.uiver.app/#q=WzAsMyxbMCwwLCJMQSJdLFsxLDAsIkQoaixBKSJdLFsyLDAsIkQoaixCKSJdLFswLDEsIlxcbGFtYmRhX3tqLEF9Il0sWzEsMiwiRChqLGYpIl1d
+\[\begin{tikzcd}
+	LA & {D(j,A)} & {D(j,B)}
+	\arrow["{\lambda_{j,A}}", from=1-1, to=1-2]
+	\arrow["{D(j,f)}", from=1-2, to=1-3]
+\end{tikzcd}\]
+    form a cone over \( D(-, B) \), and so factor uniquely through its limit \( LB \).
+    Thus we obtain \( Lf : LA \to LB \).
+    This is functorial because \( Lf \) is unique with this property.
+    This is the unique lifting of \( (LA)_{A \in \ob \mathcal C} \) to an object of \( [\mathcal C, \mathcal D] \) which makes the \( \lambda_{j,-} \) into natural transformations.
+    It is a limit cone in \( [\mathcal C, \mathcal D] \): given any cone in \( [\mathcal C, \mathcal D] \) with apex \( M \) and legs \( (\mu_{j,-})_{j \in \ob J} \) over \( D \), the \( \mu_{j,A} \) form a cone over \( D(-, A) \), so we obtain a unique \( \nu_A : MA \to LA \) such that \( \lambda_{j,A} \nu_A = \mu_{j,A} \) for all \( A \).
+    The \( \nu_A \) form a natural transformation \( M \to L \), because for any \( f : A \to B \) in \( \mathcal C \), the two paths \( \nu_B(Mf), (Lf)\nu_A : MA \rightrightarrows LB \) are factorisations of the same cone over \( D(-, B) \) through its limit, so must be equal.
+\end{proof}
+\begin{remark}
+    Note that \( f : A \to B \) is monic if and only if
+\[\begin{tikzcd}
+	A & A \\
+	A & B
+	\arrow["{1_A}", from=1-1, to=1-2]
+	\arrow["f", from=1-2, to=2-2]
+	\arrow["{1_A}"', from=1-1, to=2-1]
+	\arrow["f"', from=2-1, to=2-2]
+\end{tikzcd}\]
+    is a pullback square.
+    Thus, if \( \mathcal D \) has pullbacks, any monomorphism in \( [\mathcal C, \mathcal D] \) is a pointwise monomorphism, because the pullback in \( [\mathcal C, \mathcal D] \) is constructed pointwise by the previous lemma.
+\end{remark}
+
+\subsection{Adjoint functor theorems}
+\begin{lemma}
+    Let \( G : \mathcal D \to \mathcal C \) be a functor with a left adjoint.
+    Then \( G \) preserves all limits which exist in \( \mathcal D \).
+\end{lemma}
+\begin{proof}[Proof 1]
+    In this proof, we will assume that \( \mathcal C, \mathcal D \) both have all limits of shape \( J \).
+    If \( F \dashv G \), then the diagram
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXG1hdGhjYWwgQyJdLFsxLDAsIlxcbWF0aGNhbCBEIl0sWzAsMSwiW0osIFxcbWF0aGNhbCBDXSJdLFsxLDEsIltKLCBcXG1hdGhjYWwgRF0iXSxbMCwxLCJGIl0sWzAsMiwiXFxEZWx0YSIsMl0sWzEsMywiXFxEZWx0YSJdLFsyLDMsIltKLEZdIiwyXV0=
+\[\begin{tikzcd}
+	{\mathcal C} & {\mathcal D} \\
+	{[J, \mathcal C]} & {[J, \mathcal D]}
+	\arrow["F", from=1-1, to=1-2]
+	\arrow["\Delta"', from=1-1, to=2-1]
+	\arrow["\Delta", from=1-2, to=2-2]
+	\arrow["{[J,F]}"', from=2-1, to=2-2]
+\end{tikzcd}\]
+    commutes.
+    All of the functors in this diagram have right adjoints, so the diagram
+    % https://q.uiver.app/#q=WzAsNCxbMSwxLCJbSiwgXFxtYXRoY2FsIERdIl0sWzEsMCwiXFxtYXRoY2FsIEQiXSxbMCwwLCJcXG1hdGhjYWwgQyJdLFswLDEsIltKLCBcXG1hdGhjYWwgQ10iXSxbMCwxLCJcXGxpbV9KIiwyXSxbMSwyLCJHIiwyXSxbMCwzLCJbSiwgR10iXSxbMywyLCJcXGxpbV9KIl1d
+\[\begin{tikzcd}
+	{\mathcal C} & {\mathcal D} \\
+	{[J, \mathcal C]} & {[J, \mathcal D]}
+	\arrow["{\lim_J}"', from=2-2, to=1-2]
+	\arrow["G"', from=1-2, to=1-1]
+	\arrow["{[J, G]}", from=2-2, to=2-1]
+	\arrow["{\lim_J}", from=2-1, to=1-1]
+\end{tikzcd}\]
+    commutes up to natural isomorphism, where \( \lim_J \) sends a diagram of shape \( J \) to the apex of its limit cone.
+    But this is exactly the statement that \( G \) preserves limits.
+\end{proof}
+\begin{proof}[Proof 2]
+    In this proof, we will not assume that \( \mathcal C \) has limits of any kind, and only assume a single diagram \( D : J \to \mathcal D \) has a limit cone \( (L, (\lambda_j)_{j \in \ob J}) \) over it.
+    Given any cone over \( GD \) with apex \( A \) and legs \( \mu_j : A \to GD(j) \), the legs correspond under the adjunction to morphisms \( \overline \mu_j : FA \to D(j) \), which form a cone over \( D \) by naturality of the adjunction.
+    We obtain a unique factorisation \( \overline \mu : FA \to L \) with \( \lambda_j \overline \mu = \overline \mu_j \) for all \( j \), or equivalently, \( (G\lambda_j)\mu = \mu_j \), where \( \mu : A \to GL \) corresponds to \( \overline \mu \) under the adjunction.
+\end{proof}
+Suppose that \( \mathcal D \) has and \( G : \mathcal D \to \mathcal C \) preserves all limits.
+We would expect \( G \) to have a left adjoint.
+\begin{lemma}
+    Suppose that \( \mathcal D \) has and \( G : \mathcal D \to \mathcal C \) preserves limits of shape \( J \).
+    Then for any \( A \in \ob \mathcal C \), the category \( (A \downarrow G) \) has limits of shape \( J \), and the forgetful functor \( U : (A \downarrow G) \to \mathcal D \) creates them.
+\end{lemma}
+\begin{proof}
+    Let \( D : J \to (A \downarrow G) \) be a diagram.
+    We write each \( D(j) \) as \( (UD(j), f_j) \) where \( f_j : A \to GUD(j) \).
+    Let \( (L, (\lambda_j)_{j \in \ob J}) \) be a limit for \( UD \) in \( \mathcal D \).
+    By assumption, \( (GL, (G\lambda_j)_{j \in \ob J}) \) is a limit for \( GUD \) in \( \mathcal C \).
+    But the edges of \( \mathcal D \) are morphisms in \( (A \downarrow G) \), so the \( f_j \) form a cone over \( GUD \).
+    Thus, we obtain a unique factorisation \( f : A \to GL \) such that \( (G\lambda_j) f = f_j \) for all \( j \).
+    In other words, we have a unique lifting of \( L \) to an object \( (L, f) \) of \( (A \downarrow G) \) which makes the \( \lambda_j \) into a cone over \( D \) with apex \( (L, f) \).
+    Any cone over \( \mathcal D \) with apex \( (M, g) \), becomes a cone over \( UD \) with apex \( M \) by forgetting the structure map, so we get a unique \( h : M \to L \), and this becomes a morphism in \( (A \downarrow G) \) as both \( (Gh)g \) and \( f \) are factorisations through \( L \) of the same cone over \( UD \). 
+\end{proof}
+\begin{lemma}
+    Let \( \mathcal C \) be a category.
+    Specifying an initial object of \( \mathcal C \) is equivalent to specifying a limit for the identity functor \( 1_{\mathcal C} : \mathcal C \to \mathcal C \), considered as a diagram of shape \( \mathcal C \) in \( \mathcal C \).
+\end{lemma}
+\begin{proof}
+    First, suppose we have an initial object \( I \) in \( \mathcal C \).
+    Then the unique morphisms \( I \to A \) form a cone over \( 1_{\mathcal C} \), and it is a limit, because for any other cone \( (B, (\lambda_A : B \to A)) \), then \( \lambda_I \) is the unique factorisation as required.
+    Conversely, suppose \( (I, (\lambda_A : I \to A)) \) is a limit for \( 1_{\mathcal C} \).
+    Then certainly \( I \) is \emph{weakly initial}: it has at least one morphism to any other object, given by \( \lambda_A \).
+    For any morphism \( f : I \to A \), it is an edge of the diagram, so \( f \lambda_I = \lambda_A \), so it suffices to show that \( \lambda_I \) is the identity morphism.
+    Using the same equation with \( f = \lambda_A \), we obtain \( \lambda_A \lambda_I = \lambda_A \), so \( \lambda_I \) is a factorisation of the limit cone through itself.
+    As this factorisation must be unique, we must have \( \lambda_I = 1_I \).
+\end{proof}
+\begin{proposition}[primitive adjoint functor theorem]
+    If \( \mathcal D \) has and \( G : \mathcal D \to \mathcal C \) preserves all limits, then \( G \) has a left adjoint.
+\end{proposition}
+\begin{proof}
+    The categories \( (A \downarrow G) \) have all limits, and in particular they have initial objects, so \( G \) has a left adjoint.
+\end{proof}