diff --git a/ws2022/rav/lecture/lecture.cls b/ws2022/rav/lecture/lecture.cls index 4e8abfa..65b376d 100644 --- a/ws2022/rav/lecture/lecture.cls +++ b/ws2022/rav/lecture/lecture.cls @@ -245,7 +245,7 @@ } % replace all relations with align characters (&) and add the needed padding \regex_replace_all:nnN - { (\c{leq}&|&\c{leq}|\c{geq}&|&\c{geq}|\c{iff}&|&\c{iff}|\c{impliedby}&|&\c{impliedby}|\c{implies}&|&\c{implies}|\c{approx}&|&\c{approx}|\c{equiv}&|&\c{equiv}|=&|&=|\c{le}&|&\c{le}|\c{ge}&|&\c{ge}|&\c{stackrel}(\[.*?\])?{.*?}{.*?}|\c{stackrel}(\[.*?\])?{.*?}{.*?}&|&\c{neq}|\c{neq}&|>&|&>|<&|&<) } + { (\c{simeq}&|&\c{simeq}|\c{leq}&|&\c{leq}|\c{geq}&|&\c{geq}|\c{iff}&|&\c{iff}|\c{impliedby}&|&\c{impliedby}|\c{implies}&|&\c{implies}|\c{approx}&|&\c{approx}|\c{equiv}&|&\c{equiv}|=&|&=|\c{le}&|&\c{le}|\c{ge}&|&\c{ge}|&\c{stackrel}(\[.*?\])?{.*?}{.*?}|\c{stackrel}(\[.*?\])?{.*?}{.*?}&|&\c{neq}|\c{neq}&|>&|&>|<&|&<) } { \c{kern} \u{l_tmp_dim_needed} \1 \c{kern} \u{l_tmp_dim_needed} } \l__lec_text_tl \l__lec_text_tl diff --git a/ws2022/rav/lecture/rav.pdf b/ws2022/rav/lecture/rav.pdf index 139345e..7f65a1c 100644 Binary files a/ws2022/rav/lecture/rav.pdf and b/ws2022/rav/lecture/rav.pdf differ diff --git a/ws2022/rav/lecture/rav.tex b/ws2022/rav/lecture/rav.tex index bf79c9d..9a60048 100644 --- a/ws2022/rav/lecture/rav.tex +++ b/ws2022/rav/lecture/rav.tex @@ -30,5 +30,6 @@ Christian Merten (\href{mailto:cmerten@mathi.uni-heidelberg.de}{cmerten@mathi.un \input{rav8.tex} \input{rav9.tex} \input{rav10.tex} +\input{rav15.tex} \end{document} diff --git a/ws2022/rav/lecture/rav15.pdf b/ws2022/rav/lecture/rav15.pdf new file mode 100644 index 0000000..560de0a Binary files /dev/null and b/ws2022/rav/lecture/rav15.pdf differ diff --git a/ws2022/rav/lecture/rav15.tex b/ws2022/rav/lecture/rav15.tex new file mode 100644 index 0000000..f573b6d --- /dev/null +++ b/ws2022/rav/lecture/rav15.tex @@ -0,0 +1,311 @@ +\documentclass{lecture} + +\begin{document} + +\usetikzlibrary{shapes.misc} +\tikzset{cross/.style={cross out, draw=black, minimum size=2*(#1-\pgflinewidth), inner sep=0pt, outer sep=0pt}, +%default radius will be 1pt. +cross/.default={1pt}} + +\chapter{Hilbert's Nullstellensatz and applications} + +\section{Fields of definition} + +When $k$ is an algebraically closed field, Hilbert's Nullstellensatz gives us a bijection +between algebraic subsets of $k^{n}$ and radical ideals in $k[T_1, \ldots, T_n]$. + +This correspondence induces an anti-equivalence of categories +\begin{salign*} + \{\text{affine } k\text{-varieties}\} &\longleftrightarrow + \{\text{finitely-generated reduced } k \text{-algebras}\} \\ + (X, \mathcal{O}_X) &\longmapsto \mathcal{O}_X(X) \\ + \hat{A} = \operatorname{Hom}_{k\mathrm{-alg}}(A, k) &\longmapsfrom A +.\end{salign*} + +\begin{lemma} + Let $k$ be algebraically closed and $A$ a finitely-generated $k$-Algebra. Then + the map + \begin{salign*} + \hat{A} = \operatorname{Hom}_{k\text{-alg}}(A, k) &\longrightarrow \operatorname{Spm } A \\ + \xi &\longmapsto \text{ker } \xi + \end{salign*} + is a bijection. +\end{lemma} + +\begin{proof} + The map + admits an inverse + \begin{salign*} + \operatorname{Spm } A &\longrightarrow \operatorname{Hom}_{k\text{-alg}}(A, k) \\ + \mathfrak{m} &\longmapsto (A \to A / \mathfrak{m}) + .\end{salign*} + This is well-defined, since $A / \mathfrak{m}$ is a finite extension of the algebraically closed field + $k$, so $k \simeq A / \mathfrak{m}$. +\end{proof} + +Since we have defined a product on the left-hand side of the anti-equivalence, this must correspond +to coproduct on the right-hand side. Since the coproduct in the category of commutative +$k$-algebras with unit is given by the tensor product, we have +\[ +\mathcal{O}_{X \times Y} (X \times Y) \simeq \mathcal{O}_X(X) \otimes_k \mathcal{O}_Y(Y) +.\] + +\begin{korollar} + Let $k$ be algebraically closed. Then the tensor product of two + reduced (resp. integral) finitely-generated $k$-algebras is reduced (resp. integral). + \label{kor:k-alg-closed-tensor-of-reduced} +\end{korollar} + +\begin{proof} + This follows from the anti-equivalence of categories: Reduced since products of affine + $k$-varieties exist and integral since the product of two irreducible affine $k$-varieties is irreducible. +\end{proof} + +\begin{bem} + \ref{kor:k-alg-closed-tensor-of-reduced} is false in general if $k = \overline{k}$. For instance + $\mathbb{C}$ is an integral $\R$-algebra, but + \begin{salign*} + \mathbb{C} \otimes_{\R} \mathbb{C} + &= \R[x]/(x^2 + 1) \otimes_{\R} \mathbb{C} \\ + &= \mathbb{C}[x]/(x^2 + 1) \\ + &= \mathbb{C}[x]/((x-i)(x+i)) \\ + &\stackrel{(*)}{\simeq} \mathbb{C}[x]/(x-i) \times \mathbb{C}[x]/(x+i) \\ + &\simeq \mathbb{C} \times \mathbb{C} + \end{salign*} + is not integral, where $(*)$ follows from the Chinese remainder theorem. + + For a non-reduced example, consider $k = \mathbb{F}_{p}(t)$ and choose a $p$-th root + $\alpha = t ^{\frac{1}{p}}$ in $\overline{\mathbb{F}_p(t)}$. Then $\alpha \not\in k$ + but $\alpha ^{n} \in k$. If we put $L = k(\alpha)$, then + $\alpha \otimes 1 - 1 \otimes \alpha \neq 0$ in $L \otimes_k L$ since + the elements $(\alpha ^{i} \otimes \alpha ^{j})_{0 \le i, j \le p-1}$ form a basis + of $L \otimes_k L$ as a $k$-vector space, but + \[ + (\alpha \otimes 1 - 1 \otimes \alpha)^{p} + = \alpha ^{p} \otimes 1 - 1 \otimes \alpha ^{p} + = 1 \otimes \alpha ^{p} - 1 \otimes \alpha ^{p} = 0 + .\] +\end{bem} + +We now consider more generally finitely generated reduced $k$-algebras when $k$ is not +necessarily closed. + +\begin{bsp} + Let $A = \R[X]/(x^2 +1)$. Since $x^2 + 1 $ is irreducible in $\R[x]$, it + generates a maximal ideal, thus the finitely-generated $\R$-algebra $A$ is a field and in + particular reduced. We can equip the topogical space + $X \coloneqq \operatorname{Spm } A = \{ (0)\} $ with a sheaf of regular functions, defined + by $\mathcal{O}_X(\{(0)\}) = A$. In other words, $\operatorname{Spm } A$ is just a point, + but equipped with the reduced $\R$-algebra $A$. It thus differs from the + point $\operatorname{Spm } \R$, which is equipped with the reduced $\R$-algebra $\R$, + since $\R[x]/(x^2 + 1) \not\simeq \R$ as $\R$-algebras. Indeed, the $\R$-algebra $\R[x]/(x^2+1)$ + is $2$ dimensional as a real vector space. + + $A$ possesses a non-trivial $\R$-algebra automorphism induced by the automorphism of $\R$-algebras, + $P \mapsto P(-x)$ in $\R[x]$. Indeed, $\R[x]/(x^2+1) \simeq \mathbb{C}$ as $\R$-algebras, + with the previous automorphism corresponding to the complex conjugation $z \mapsto \overline{z}$. +\end{bsp} + +\begin{bsp} + By analogy with the Zariski topology on maximal spectra of (finitely generated, reduced) + $\mathbb{C}$-algebras, we can equip $X = \operatorname{Spm } A$ with + a Zariski topology for all (finitely generated reduced) $\R$-algebras $A$: the closed subsets + of this topology are given by + \[ + \mathcal{V}_X(I) \coloneqq \{ \mathfrak{m} \in \operatorname{Spm } A \mid \mathfrak{m} \supset I\} + \] for any ideal $I \subseteq A$. + Note that $X = \operatorname{Spm } A$ contains + $\hat{A} = \operatorname{Hom}_{k\text{-alg}}(A, k)$ as a subset: the points + of $\hat{A}$ correspond to maximal ideals $\mathfrak{m}$ of $A$ with residue field + $A / \mathfrak{m} \simeq k$. But when $k \not\simeq \overline{k}$, the set + $\operatorname{Spm } A$ is strictly larger than $\hat{A}$: it contains maximal ideals $\mathfrak{m}$ + such that $A / \mathfrak{m}$ is a non-trivial finite extension of $k$. The induced topology on + $\hat{A} \subseteq \operatorname{Spm } A$ is the Zariski topologoy of $\hat{A}$ that was + introduced earlier. + + Let $A = \R[x]$. Maximal ideals in the principal ring $\R[x]$ are generated + by a single irreducible polynomial $P$, which is either of degree $1$ or of degree $2$ with + negative discriminant. + + In the first case, $P = x-a$ for some $a \in \R$ and the residue field is $\R[x]/(x - a) \simeq \R$, + while, in the second case, $P = x^2 + bx + c$ for $b, c \in \R$ and $b^2 - 4c < 0$ and + by choosing a root $z_0$ of $P$ in $\mathbb{C}$, the map + \begin{salign*} + \eta_{z_0} \colon \R[x]/(x^2 + bx + c) &\longrightarrow \mathbb{C} \\ + \overline{P} &\longmapsto P(z_0) + \end{salign*} + is a field-homomorphism. In particular it is injective. Since $\mathbb{C}$ and + $\R[x]/(x^2 + bx + c)$ are both degree $2$ extensions of $\R$, we have + $\R[x]/(x^2 + bx + c) \simeq \mathbb{C}$. + Note that the other root of $x^2 + bx +c $ is $\overline{z_0}$ and that + $\eta_{\overline{z_0}} = \sigma \circ \eta_{z_0}$ where $\sigma$ is complex conjugation on $\mathbb{C}$. + So we have to ways to identify $\R[x]/(x^2 + bx +c)$ to $\mathbb{C}$ and they are + related by the action of $\text{Gal}(\mathbb{C}/ \R)$ on $\mathbb{C}$. + + To sum up, the difference between the two possible types of maximal ideals $\mathfrak{m} \subseteq \R[x]$ + is the residue field, which is either $\R$ or $\mathbb{C}$. When it is $\R$, we + find exactly the points of + \begin{salign*} + \widehat{\R[x]} &= \operatorname{Hom}_{\R\text{-alg}}(\R[x], \R) \\ + &\simeq \{ \mathfrak{m} \in \operatorname{Spm } \R[x] \mid \R[x]/\mathfrak{m} \simeq \R\} \\ + &\simeq \{ (x-a) \colon a \in \R\} \\ + &\simeq \R + .\end{salign*} + And when the residue field is $\mathbb{C}$, we have $\mathfrak{m} = (x^2 + bx + c)$ with + $b, c \in \R$ such that $b^2 - 4c < 0$. If we choose $z_0$ to be the root + of $x^2 + bx +c$ with $\text{Im}(z_0) > 0$, we can identify the set of these maximal ideals with + the subset + \[ + H \coloneqq \{ z \in \mathbb{C} \mid \text{Im}(z) > 0\} + .\] + In other words, the following pictures emerges, where we identify + $\operatorname{Spm } \R[x]$ with + \[ + \hat{H} \coloneqq \{ z \in \mathbb{C} \mid \text{Im}(z) \ge 0\} + \] + via the map + \begin{salign*} + \operatorname{Spm } \R[x] &\longrightarrow \hat{H} \\ + \mathfrak{m} &\longmapsto \begin{cases} + a \in \R & \mathfrak{m} = (x-a) \\ + z_0 \in H & \mathfrak{m} = ((x-z_0)(x-\overline{z_0})) \text{ and } \text{Im}(z_0) > 0 + \end{cases} + \end{salign*} + which is indeed bijective. + %\begin{figure} + % \centering + % \begin{tikzpicture} + % \draw[red] (-2, 0) -- (2,0) node[right] {$\R \simeq \operatorname{Hom}_{\R\text{-alg}}(\R[x], \R)$}; + % \draw[->] (0, 0) -- (0,4); + % \end{tikzpicture} + % \caption{$\operatorname{Spm } \R[x] \simeq \hat{H} + % = \left\{ z \in \mathbb{C} : \text{Im}(z) \ge 0 \right\}$} + %\end{figure} + We see that $\operatorname{Spm } \R[x]$ contains a lot more points + that $\R$. One could go further and add the ideal $(0)$: This would give the set + \[ + \mathbb{A}^{1}_{\R} = \operatorname{Spec } \R[x] + = \operatorname{Spm } \R[x] \cup \{(0)\} + .\] +\end{bsp} + +\begin{bem} + If $A$ is a $k$-algebra and $\overline{k}$ is an algebraic closure of $k$, the + group $\text{Aut}_k(\overline{k})$ acts on the $\overline{k}$-algebra + $A_{\overline{k}} \coloneqq A \otimes_k \overline{k}$ via + $\sigma (a \otimes \lambda) \coloneqq a \otimes \sigma(\lambda)$. Moreover, the map + $a \mapsto a \otimes 1$ induces an injective morphism of $k$-algebras + $A \xhookrightarrow{} A \otimes_k \overline{k}$ since + the tensor product over fields is left-exact. + Its image is contained in the $k$-subalgebra + $\operatorname{Fix}_{\operatorname{Aut}_k(\overline{k})} A_{\overline{k}} \subseteq A_{\overline{k}}$. When + $k$ is a perfect field, this inclusion is an equality. +\end{bem} + +\begin{bsp} + If $A = \R[x]$, then $A \otimes_{\R} \mathbb{C} \simeq \mathbb{C}[x]$. The group + $\text{Aut}_{\R}(\mathbb{C}) = \text{Gal}(\mathbb{C}/\R) = \langle \sigma \rangle$ with + $\sigma\colon z \mapsto \overline{z}$, acts naturally on $\mathbb{C}[x]$. This + is an action by $\R$-algebra automorphisms. Clearly, + $\text{Fix}_{\langle\sigma\rangle} \mathbb{C}[x] = \R[x]$. There + is an induced action on $\operatorname{Spm } \mathbb{C}[x]$, + defined by + \[ + \sigma(\mathfrak{m}) = \sigma((x-z)) \coloneqq (x - \sigma(z)) = (x - \overline{z}) + .\] + When we identify $\operatorname{Spm } \mathbb{C}[x]$ with $\mathbb{C}$ + via $(x-z) \mapsto z$, this action is just $z \mapsto \overline{z}$. This + ,,geometric action'' induces an action of $\text{Gal}(\mathbb{C}/\R)$ on + regular functions on $\mathbb{C}$: to $h \in \mathcal{O}_{\mathbb{C}}(U)$, there + is associated a regular function $h \in \mathcal{O}_{\mathbb{C}}(\sigma(U))$, defined for + all $x \in \sigma(U)$, by + \[ + \sigma(h)(z) \coloneqq \sigma \circ h \circ \sigma ^{-1}(z) = \overline{h(\overline{z})} + .\] + In particular, if $h = P \in \mathcal{O}_{\mathbb{C}}(\mathbb{C}) = \mathbb{C}[x]$, then + $P \mapsto \sigma(P)$ coincides with the natural + $\text{Gal}(\mathbb{C} / \R)$ action on $\mathbb{C}[x]$. We will see momentarily that this + defines a sheaf of $\R$-algebras on $\operatorname{Spm } \R[x]$. To that end, + let us first look more closely at the $\text{Gal}(\mathbb{C}/ \R)$ action + on $\operatorname{Spm } \mathbb{C}[x]$. Its fixed-point set + is + \[ + \{ \mathfrak{m} \in \operatorname{Spm } \mathbb{C}[x] \mid \mathfrak{m} = (x-a), a \in \R\} + \simeq \R = \operatorname{Fix}_{z \mapsto \overline{z}}(\mathbb{C}) + .\] + Moreover, there is a map + \begin{salign*} + \operatorname{Spm } \mathbb{C}[x] &\longrightarrow \operatorname{Spm } \R[x] \\ + \mathfrak{m} &\longmapsto \mathfrak{m} \cap \R[x] + \end{salign*} + sending $(x-a) \mathbb{C}[x]$ to $(x-a)\R[x]$ if $a \in \R$, + and $(x-z)\mathbb{C}[x]$ to $(x-z)(x-\overline{z})\R[x]$ if $z \in \mathbb{C} \setminus \R$. + This map is surjective and induces a bijection + \[ + (\operatorname{Spm } \mathbb{C}[x]) / \operatorname{Gal}(\mathbb{C} / \R) + \xlongrightarrow{\simeq} \operatorname{Spm } \R[x] + .\] + Geometrically, the quotient map $\pi\colon \operatorname{Spm } \mathbb{C}[x] \to \operatorname{Spm } \R[x]$ + is the ,,folding map`` + \begin{salign*} + \mathbb{C} &\longrightarrow \hat{H} \\ + z = u + iv &\longmapsto u + i |v| + .\end{salign*} + \begin{figure} + \centering + \begin{tikzpicture} + \draw[red] (-2, 0) -- (2,0) node[right] {$\R$}; + \fill (1, -1) circle[radius=0.75pt] node[right] {$z_0$}; + \draw[->] (0,-1.5) -- (0,2); + \draw[->] (3.2,0) -- node[above] {$\pi$} (4.2,0); + \draw[red] (5, 0) -- (9,0) node[right] {$\R$}; + \fill (8, 1) circle[radius=0.75pt] node[right] {$\pi(z_0)$}; + \draw[->] (7, 0) -- (7,2); + \end{tikzpicture} + \caption{The quotient map + $\pi\colon \operatorname{Spm } \mathbb{C}[x] \to \operatorname{Spm } \R[x]$ is geometrically a folding.} + \end{figure} + In view of this, it is natural to + \begin{enumerate}[(i)] + \item put the quotient topology on + \[ + \operatorname{Spm } \R[x] = \left( \operatorname{Spm } \mathbb{C}[x] \right) + / \operatorname{Gal}(\mathbb{C}/\R) + \] + where $\operatorname{Spm } \mathbb{C}[x] \simeq \mathbb{C}$ is equipped with its topology + of algebraic variety. + \item define a sheaf of $\R$-algebras on $\operatorname{Spm } \R[x]$ by pushing-forward + the structure sheaf on $\operatorname{Spm } \mathbb{C}[x]$ + and then taking the $\operatorname{Gal}(\mathbb{C}/ \R)$-invariant subsheaf: + \[ + \mathcal{O}_{\operatorname{Spm } \R[x]}(U) + \coloneqq \mathcal{O}_{\operatorname{Spm } \mathbb{C}[x]} + (\pi^{-1}(U))^{\operatorname{Gal}(\mathbb{C} / \R)} + \] where + $\pi\colon \operatorname{Spm } \mathbb{C}[x] \to \operatorname{Spm } \R[x]$, + $\mathfrak{m} \mapsto \mathfrak{m} \cap \R[x]$ is the quotient map, + and $\operatorname{Gal}(\mathbb{C}/ \R)$ acts on + $\mathcal{O}_{\operatorname{Spm } \mathbb{C}[x]}(\pi^{-1}(U))$ via + $h \mapsto \sigma(h) = \sigma \circ h \circ \sigma ^{-1}$ (note that the open set + $\pi^{-1}(U)$ is $\operatorname{Gal}(\mathbb{C} / \R)$-invariant). + \end{enumerate} + Observe that + \[ + \mathcal{O}_{\operatorname{Spm } \R[x]}(\operatorname{Spm } \R[x]) + = \mathbb{C}[x]^{\operatorname{Gal}(\mathbb{C} / \R)} = \R[x] + .\] + Also, if $h = \frac{f}{g}$ around $x \in U$, then, around + $\sigma(x) \in U$, one has $\sigma(h) = \frac{\sigma(f)}{\sigma(g)}$ and, + for all $\lambda \in \mathbb{C}$, $\sigma(\lambda h) = \overline{\lambda} \sigma(h)$. + + Remarkably, we will see that we can reconstruct the algebraic $\mathbb{C}$-variety + \[ + (X_{\mathbb{C}}, \mathcal{O}_{X_{\mathbb{C}}}) + \coloneqq (\operatorname{Spm } \mathbb{C}[x], \mathcal{O}_{\operatorname{Spm } \mathbb{C}[x]}) + \] from the ringed space + \[ + (X, \mathcal{O}_X) \coloneqq (\operatorname{Spm } \R[x], \mathcal{O}_{\operatorname{Spm } \R[x]} + \] that we have just constructed. +\end{bsp} + +\end{document} diff --git a/ws2022/rav/lecture/rav5.tex b/ws2022/rav/lecture/rav5.tex index ba63a41..2ab7c12 100644 --- a/ws2022/rav/lecture/rav5.tex +++ b/ws2022/rav/lecture/rav5.tex @@ -2,7 +2,7 @@ \begin{document} -\chapter{Affine varieties} +\chapter{Algebraic varieties} \section{Spaces with functions}