Lectures 10B

iii/alggeom/03_schemes.tex

@@ -149,10 +149,15 @@ \subsection{Definitions and examples}
     \[ \mathcal O_U(U) = k[x,x^{-1},y] \cap k[x,y,y^{-1}] \subseteq k[x,x^{-1},y,y^{-1}] \]
     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.
+
+    In general, if \( X \) is a scheme, \( f \in \Gamma(X, \mathcal O_X) = \mathcal O_X(X) \), and \( p \in X \), then there is a well-defined stalk \( \mathcal O_{X,p} \) at \( p \), which is of the form \( A_{\mathfrak p} \) up to isomorphism, where \( \mathfrak p \) is a prime ideal.
+    To say this, we are using an isomorphism of an open set \( V_p \) containing \( p \) to \( \Spec A \).
+    In particular, \( A_{\mathfrak p} \) has a unique maximal ideal, namely \( \mathfrak p A_{\mathfrak p} \).
+    We say that \( f \) vanishes at \( p \) if its image in \( \faktor{A_{\mathfrak p}}{\mathfrak p A_{\mathfrak p}} \), or equivalently, \( f \in \mathfrak p A_{\mathfrak p} \).
+    As a consequence, the vanishing locus \( \mathbb V(f) \subseteq X \) is well-defined.
 \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
 \[ \varphi_{\alpha\beta} : \eval{\mathcal F_\alpha}_{U_\alpha \cap U_\beta} \to \eval{\mathcal F_\beta}_{U_\alpha \cap U_\beta} \]
@@ -181,10 +186,78 @@ \subsection{Gluing sheaves}
     \[ \varphi_{\alpha\beta} \circ \varphi_{\gamma\alpha}\qty(\eval{s}_{V \cap U_\alpha \cap U_\beta}) = \varphi_{\gamma\beta}\qty(\eval{s}_{V \cap U_\alpha \cap U_\beta}) \]
 \end{proof}
 
-\subsection{???}
+\subsection{Gluing schemes}
+Let \( (X, \mathcal O_X) \) and \( (Y, \mathcal O_Y) \) be schemes with open sets \( U \subseteq X, V \subseteq Y \), and let \( \varphi : (U, \eval{\mathcal O_X}_U) \to (V, \eval{\mathcal O_Y}_V) \) be an isomorphism.
+The topological spaces \( X, Y \) can be glued on \( U, V \) using \( \varphi \).
+
+First, take \( S = \faktor{X \sqcup Y}{U \sim V} \).
+By definition of the quotient topology, the images of \( X \) and \( Y \) in \( S \) form an open cover, and their intersection is the image of \( U \), or equivalently, the image of \( V \).
+Now, we can glue the structure sheaves on these open sets as described in the previous subsection.
+Note that in this case, there is no cocycle condition.
+\begin{example}[the bug-eyed line; the line with doubled origin]
+    Let \( k \) be a field.
+    Let \( X = \Spec k[t] \) and \( Y = \Spec k[u] \).
+    Let
+    \[ U = \Spec k[t, t^{-1}] = \Spec k[t]_t = U_t \subseteq X;\quad V = \Spec k[u, u^{-1}] = \Spec k[u]_u = U_u \subseteq Y \]
+    We define the isomorphism \( \varphi : U \to V \) given by \( t \mapsfrom u \).
+    Technically, we define an isomorphism of rings \( k[u, u^{-1}] \to k[t, t^{-1}] \) by \( u \mapsto t \) and then apply \( \Spec \).
+    At the level of topological spaces, \( X = \mathbb A^1_k \) and \( Y = \mathbb A^1_k \), so \( U = \mathbb A^1_k \setminus \qty{(t)} \) and \( V = \mathbb A^1_k \setminus \qty{(u)} \).
+    Gluing along this isomorphism, we obtain a scheme \( S \) which is a copy of \( \mathbb A^1_k \) but with two origins.
+    Note that the generic points in \( X \) and \( Y \) lie in \( U \) and \( V \) respectively, and thus are glued into a single generic point in \( S \).
+
+    Consider the open sets in \( S \).
+    Open sets entirely contained within \( X \) and \( Y \) yield open sets in \( S \).
+    We also have open sets of the form \( W = S \setminus \qty{\mathfrak p_1, \dots, \mathfrak p_r} \) where \( \mathfrak p_i \) is contained in \( U \) or \( V \).
+    One example is \( W = S \); we can calculate \( \mathcal O_S(S) \) using the sheaf axioms, and one can show that it is isomorphic to \( k[t] \).
+    We can conclude that \( S \) is not an affine scheme, because there is a maximal ideal in \( k[t] \) where the vanishing locus is precisely two points.
+\end{example}
+\begin{example}[the projective line]
+    Let \( X = \Spec k[t] \) and \( Y = \Spec k[s] \), and define \( U = \Spec k[t,t^{-1}], V = \Spec k[s,s^{-1}] \) as above.
+    We glue these schemes using the isomorphism \( s \mapsto t^{-1} \), giving the projective line \( \mathbb P^1_k \).
+\end{example}
+\begin{proposition}
+    \( \mathcal O_{\mathbb P^1_k}(\mathbb P^1_k) = k \).
+\end{proposition}
+% this does not require that k is algebraically closed
+\begin{proof}[Proof sketch]
+    We use the same idea as in the previous example.
+    The only elements of \( k[t, t^{-1}] \) that are both polynomials in \( t \) and \( t^{-1} \) are the constants.
+    % important exercise.
+\end{proof}
+In particular, \( \mathbb P^1_k \) is not an affine scheme.
+\begin{example}
+    We can similarly build a scheme \( S \) which is a copy of \( \mathbb A^2_k \) with a doubled origin.
+    This has the interesting property that there exist affine open subschemes \( U_1, U_2 \subseteq S \) such that \( U_1 \cap U_2 \) is not affine; we can take \( U_1 \) and \( U_2 \) to be \( S \) but with one of the origins deleted.
+    Note that \( \mathbb A^1_k \) without the origin is affine.
+\end{example}
+
+Let \( \qty{X_i}_{i \in I} \) be schemes, \( X_{ij} \subseteq X_i \) be open subschemes, and \( f_{ij} : X_{ij} \to X_{ji} \) be isomorphisms such that
+\[ f_{ii} = \id_{X_i};\quad f_{ij} = f_{ji}^{-1};\quad f_{ik} = f_{jk} \circ f_{ij} \]
+where the last equality holds whenever it is defined.
+Then there is a unique scheme \( X \) with an open cover by the \( X_i \), glued along these isomorphisms.
+This is an elaboration of the above construction, which is discussed on the first example sheet.
+
+Let \( A \) be a ring, and let \( X_i = \Spec A\qty[\frac{x_0}{x_i}, \dots, \frac{x_n}{x_i}] \).
+Let \( X_{ij} = \mathbb V\qty(\frac{x_j}{x_i})^c \subseteq X_i \).
+We define the isomorphisms \( X_{ij} \to X_{ji} \) by \( \frac{x_k}{x_i} \mapsto \frac{x_k}{x_j} \qty(\frac{x_i}{x_j})^{-1} \).
+The resulting glued scheme is called \emph{projective \( n \)-space}, denoted \( \mathbb P^n_A \).
+% 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
+    \[ A = \bigoplus_{i \in \mathbb Z} A_i \]
+    as abelian groups, such that \( A_i A_j \subseteq A_{i+j} \).
+\end{definition}
+\begin{example}
+    Let \( A = k[x_0, \dots, x_n] \), and let \( A_d \) be the set of degree \( d \) homogeneous polynomials, together with the zero polynomial.
+\end{example}
 \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.
+    Let \( I \subseteq k[x_0, \dots, x_n] \) be a homogeneous ideal; that is, an ideal generated by homogeneous elements of possibly different degrees.
+    Then, for \( A = k[x_0, \dots, x_n] \), the ring \( \faktor{A}{I} \) is also naturally graded.
+    % 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 \( A_0 \).

iii/cat/04_limits.tex

@@ -154,5 +154,100 @@ \subsection{Limits}
         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}.
+
+        If any category \( \mathcal C \) has binary products and equalisers, we can construct all pullbacks.
+        First, we construct the product \( A \times B \), then we form the equaliser of \( f \pi_1, g \pi_2 : A \times B \rightrightarrows C \).
+        This yields the pullback.
+        \item Let \( M \) be the two-element monoid \( \qty{1, e} \) with \( e^2 = e \).
+        A diagram of shape \( M \) in a category \( \mathcal C \) is an object of \( \mathcal C \) equipped with an idempotent endomorphism.
+        A cone over this diagram is a morphism \( f : B \to A \) such that \( ef = f \).
+        A limit (respectively colimit) is the monic (respectively epic) part of a splitting of \( e \).
+        This is because the pair \( (e, 1_A) \) has an equaliser if and only if \( e \) splits.
+        % Note that the functor \( F : \mathbf{Set} \to [M, \mathbf{Set}] \) is the constant map \( \Delta \), which explains why the left and right adjoints coincide. % what is 3.2(e)?
+        \item Let \( \mathbb N \) be the poset category of the natural numbers.
+        A diagram of shape \( \mathbb N \) is a \emph{direct sequence} of objects, which consists of objects \( A_0, A_1, \dots \) and morphisms \( f_i : A_i \to A_{i+1} \).
+        A colimit for this diagram is a \emph{direct limit}, which consists of an object \( A_\infty \) and morphisms \( g_i : A_i \to A_\infty \) which are compatible with the \( f_i \).
+        Dually, an \emph{inverse sequence} is a diagram of shape \( \mathbb N^\cop \), and a limit for this diagram is called an \emph{inverse limit}.
+        For example, an infinite-dimensional CW-complex \( X \) is the direct limit of its \( n \)-dimensional skeletons in \( \mathbf{Top} \).
+        The ring of \( p \)-adic integers is the limit of the inverse sequence defined by \( A_n = \faktor{\mathbb Z}{p^n\mathbb Z} \) in \( \mathbf{Rng} \).
+    \end{enumerate}
+\end{example}
+\begin{lemma}
+    Let \( \mathcal C \) be a category.
+    \begin{enumerate}
+        \item If \( \mathcal C \) has equalisers and all small products, then \( \mathcal C \) has all small limits.
+        \item If \( \mathcal C \) has equalisers and all finite products, then \( \mathcal C \) has all finite limits.
+        \item If \( \mathcal C \) has pullbacks and a terminal object, then \( \mathcal C \) has all finite limits.
+    \end{enumerate}
+\end{lemma}
+Note that the empty product is implicitly included in (i) and (ii).
+A terminal object is a product over no factors.
+\begin{proof}
+    \emph{Parts (i) and (ii).}
+    We prove (i) and (ii) in the same way.
+    Let \( D : J \to \mathcal C \) be a diagram.
+    We form the products
+    \[ P = \prod_{j \in \ob J} D(j);\quad Q = \prod_{\alpha \in \mor J} D(\cod \alpha) \]
+    These are small or finite as required.
+    We have morphisms \( f, g : P \rightrightarrows Q \) defined by
+    \[ \pi_\alpha f = \pi_{\cod \alpha};\quad \pi_\alpha g = D(\alpha) \pi_{\dom \alpha} \]
+    Let \( e : E \to P \) be an equaliser for \( f \) and \( g \), and define \( \lambda_j = \pi_j e : E \to D(j) \).
+    These morphisms form a cone, since if \( \alpha : j \to j' \), we have
+    \[ D(\alpha) \lambda_j = D(\alpha) \pi_j e = \pi_\alpha g e = \pi_\alpha f e = \pi_{j'} e = \lambda_{j'} \]
+    Given any cone \( (A, (\mu_j)_{j \in \ob J}) \) over \( D \), we have a unique \( \mu : A \to P \) with \( \pi_j \mu = \mu_j \) for all \( j \).
+    Then,
+    \[ \pi_\alpha f \mu = \mu_{\cod \alpha} = D(\alpha) \mu_{\dom \alpha} = \pi_\alpha g \mu \]
+    for all \( \alpha \), so \( \mu \) factors uniquely through \( e \).
+
+    \emph{Part (iii).}
+    We show that the hypotheses of (iii) imply those of (ii).
+    If \( 1 \) is the terminal object, we form the pullback of the span
+    % https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzEsMSwiMSJdLFswLDEsIkIiXSxbMCwxXSxbMiwxXV0=
+\[\begin{tikzcd}
+	& A \\
+	B & 1
+	\arrow[from=1-2, to=2-2]
+	\arrow[from=2-1, to=2-2]
+\end{tikzcd}\]
+    This has the universal property of the product \( A \times B \), so \( \mathcal C \) has binary products and hence all finite products by induction.
+    To construct the equaliser of \( f, g : A \rightrightarrows B \), we consider the pullback of
+    % https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzEsMSwiQSBcXHRpbWVzIEIiXSxbMCwxLCJBIl0sWzAsMSwiKDFfQSwgZikiXSxbMiwxLCIoMV9BLCBnKSIsMl1d
+\[\begin{tikzcd}
+	& A \\
+	A & {A \times B}
+	\arrow["{(1_A, f)}", from=1-2, to=2-2]
+	\arrow["{(1_A, g)}"', from=2-1, to=2-2]
+\end{tikzcd}\]
+    Any cone over this diagram has its two legs \( C \rightrightarrows A \) equal, so a pullback is an equaliser for \( f, g \).
+\end{proof}
+\begin{definition}
+    A category is called \emph{complete} if it has all small limits, and \emph{cocomplete} if it has all small colimits.
+\end{definition}
+\begin{example}
+    The categories \( \mathbf{Set}, \mathbf{Gp}, \mathbf{Top} \) are complete and cocomplete.
+\end{example}
+
+\subsection{Preservation and creation}
+\begin{definition}
+    Let \( G : \mathcal D \to \mathcal C \) be a functor.
+    We say that \( G \)
+    \begin{enumerate}
+        \item \emph{preserves} limits of shape \( J \) if whenever \( D : J \to \mathcal D \) is a diagram with limit cone \( (L, (\lambda_j)_{j \in \ob J}) \), the cone \( (GL, (G\lambda_j)_{j \in \ob J}) \) is a limit for \( GD \);
+        \item \emph{reflects} limits of shape \( J \) if whenever \( D : J \to \mathcal D \) is a diagram and \( (L, (\lambda_j)_{j \in \ob J}) \) is a cone such that \( (GL, (G\lambda_j)_{j \in \ob J}) \) is a limit for \( GD \), then \( (L, (\lambda_j)_{j \in \ob J}) \) is a limit for \( D \);
+        \item \emph{creates} limits of shape \( J \) if whenever \( D : J \to \mathcal D \) is a diagram with limit cone \( (M, (\mu_j)_{j \in \ob J}) \) for \( GD \) in \( \mathcal C \), there exists a cone \( (L, (\lambda_j)_{j \in \ob J}) \) over \( D \) such that \( (GL, (G\lambda_j)_{j \in \ob J}) \cong (M, (\mu_j)_{j \in \ob J}) \) in \( \operatorname{Cone}(GD) \), and any such cone is a limit for \( D \).
+    \end{enumerate}
+\end{definition}
+We typically assume in (i) that \( \mathcal D \) has all limits of shape \( J \), and we assume in (ii) and (iii) that \( \mathcal C \) has all limits of shape \( J \).
+With these assumptions, \( G \) creates limits of shape \( J \) if and only if \( G \) preserves and reflects limits, and \( \mathcal D \) has all limits of shape \( J \).
+\begin{corollary}
+    In any of the statements of the previous lemma, we can replace both instances of `\( \mathcal C \) has' by either `\( \mathcal D \) has and \( G : \mathcal D \to \mathcal C \) preserves' or `\( \mathcal C \) has and \( G : \mathcal D \to \mathcal C \) creates'.
+\end{corollary}
+\begin{example}
+    \begin{enumerate}
+        \item The forgetful functor \( U : \mathbf{Gp} \to \mathbf{Set} \) creates all small limits.
+        It does not preserve colimits, as in particular it does not preserve coproducts.
+        \item The forgetful functor \( U : \mathbf{Top} \to \mathbf{Set} \) preserves all small limits and colimits, but does not reflect them, as we can retopologise the apex of a limit cone.
+        \item The inclusion \( \mathbf{AbGp} \to \mathbf{Gp} \) reflects coproducts, but does not preserve them.
+        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}

iii/mtncl/02_quantifier_elimination.tex

@@ -531,5 +531,34 @@ \subsection{Interaction with other properties}
     The field of fractions of \( \mathcal C \) can be made an ordered field in a canonical way, by saying \( \frac{a}{b} > 0 \) if \( ab > 0 \).
     The embedding \( \mathcal C \) into \( \mathcal A \) is an injective homomorphism of ordered rings, into an ordered field.
     By the universal property of the fraction field, there is a unique homomorphism of ordered fields from \( FF(\mathcal C) \) to \( \mathcal A \) that extends the inclusion of \( \mathcal C \) into \( \mathcal A \).
-    Let \( \mathcal A' \) be the real closure of \( FF(\mathcal C) \).
+    Let \( \mathcal A' \) be the real closure of \( FF(\mathcal C) \), so that \( \mathcal C \subseteq FF(\mathcal C) \subseteq \mathcal A' \subseteq \mathcal A \).
+    If \( \mathcal B \vDash \mathsf{RCF} \) and \( \mathcal C \subseteq B \), then by the same argument we have a unique ordered ring homomorphism \( FF(\mathcal C) \to \mathcal B \) extending the embedding \( \mathcal C \subseteq \mathcal B \).
+    Thus \( \mathcal A' \subseteq \mathcal B \) as well, and this embedding fixes \( \mathcal C \).
 \end{example}
+\begin{corollary}[Hilbert's Nullstellensatz]
+    Let \( k \) be an algebraically closed field, and \( I \) be a proper ideal of \( k[x_1, \dots, x_n] \).
+    Then there exists \( \vb a \in k^n \) such that \( f(\vb a) = 0 \) for all \( I \in f \).
+\end{corollary}
+\begin{proof}
+    By Zorn's lemma, every proper ideal can be extended to a maximal ideal, so without loss of generality we may assume that \( I \) is a maximal ideal.
+    Let \( L \) be the residue field \( \faktor{k[x_1, \dots, x_n]}{I} \), and let \( \overline L \) be its algebraic closure.
+    By Hilbert's basis theorem, there exists a finite set of generators \( f_1, \dots, f_r \) for \( I \).
+    Note that \( \vb 0 \) is a witness to
+    \[ \overline L \vDash \exists \vb x.\, \qty(f_1(\vb x) = 0 \wedge \dots \wedge f_r(\vb x) = 0) \]
+    We have embeddings \( k \subseteq L \subseteq \overline L \), where both \( k \) and \( \overline L \) are algebraically closed fields.
+    The theory of algebraically closed fields has quantifier elimination, so is model-complete.
+    Thus the embedding \( k \subseteq \overline L \) is elementary, so
+    \[ k \vDash \exists \vb x.\, \qty(f_1(\vb x) = 0 \wedge \dots \wedge f_r(\vb x) = 0) \]
+    We can then take \( \vb a \) to be a witness to this existential.
+\end{proof}
+\begin{corollary}[Chevalley's theorem]
+    Let \( k \) be an algebraically closed field.
+    Then the image of a constructible set in \( k^n \) under a polynomial map is constructible.
+\end{corollary}
+\begin{proof}
+    The quantifier-free-definable subsets of \( k^n \) are precisely the finite Boolean combinations of the Zariski closed subsets of \( k^n \), which are by definition the constructible sets.
+    As \( \mathsf{ACF} \) has quantifier elimination, these are exactly the definable subsets using arbitrary formulae.
+    Now, if \( X \subseteq k^n \) is constructible and \( p : k^n \to k^m \) is a polynomial map, then
+    \[ p(X) = \qty{y \in k^m \mid \exists x.\, p(x) = y} \]
+    This is definable in the same language, so is a constructible set.
+\end{proof}