From 617cb787cf1b2d4a2653d532a69ff542f25879d0 Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Sun, 12 Jun 2016 00:39:55 +0300 Subject: [PATCH] Initial commit: first three lectures --- .gitignore | 12 + motives.pdf | Bin 0 -> 444581 bytes motives.tex | 1191 +++++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 1203 insertions(+) create mode 100644 .gitignore create mode 100644 motives.pdf create mode 100644 motives.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..40cb740 --- /dev/null +++ b/.gitignore @@ -0,0 +1,12 @@ +*~ +.DS_Store +*.aux +*.fdb_latexmk +*.fls +*.log +*.out +*.synctex.gz +*.toc +*.brf +*.idx +*.ilg diff --git a/motives.pdf b/motives.pdf new file mode 100644 index 0000000000000000000000000000000000000000..88e8bc4deae3508ff277c759da1300221950e58c GIT binary patch literal 444581 zcma&NLz^fJtY%xbZQHhO+qP}nwr$(CZQFL$?(g<m>S!g zm_qUKK{>lPnHt(cd2F>}PsZuAKl156`bfdE$R>~ag5VmBD;Ype6mfzixIhqTh~a=v 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+\documentclass[a4paper]{article} +\usepackage[T2A]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[russian]{babel} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{fullpage} +\usepackage{rotating} +\usepackage{stmaryrd} +\usepackage{mathtools} +\usepackage{bbm} +\usepackage[unicode,colorlinks=true,pagebackref=true]{hyperref} +\usepackage[all]{xy} +\usepackage{microtype} +\usepackage{amsthm} + +\usepackage{tikz} +\usetikzlibrary{calc} +\usetikzlibrary{through} + +\DeclareFontFamily{OT1}{pzc}{} +\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.1] pzcmi7t}{} +\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it} + +\DeclareMathOperator{\PGSp}{PGSp} +\DeclareMathOperator{\PGO}{PGO} +\DeclareMathOperator{\ind}{ind} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\Gr}{Gr} +\DeclareMathOperator{\Cent}{Cent} +\DeclareMathOperator{\Spec}{Spec} +\DeclareMathOperator{\Map}{Map} +\DeclareMathOperator{\GW}{GW} +\DeclareMathOperator{\rk}{rk} +\DeclareMathOperator{\Br}{Br} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Lie}{Lie} +\DeclareMathOperator{\PGL}{PGL} +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\SL}{SL} +\DeclareMathOperator{\fchar}{char} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\Iso}{Iso} +\DeclareMathOperator{\SB}{SB} +\DeclareMathOperator{\SO}{SO} +\DeclareMathOperator{\Spin}{Spin} +\DeclareMathOperator{\Isom}{Isom} +\DeclareMathOperator{\im}{im} +\DeclareMathOperator{\disc}{disc} +\DeclareMathOperator{\Stab}{Stab} +\DeclareMathOperator{\Nrd}{Nrd} + +%\DeclareFontFamily{OT1}{pzc}{} +%\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.2] pzcmi7t}{} +%\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it} +\newcommand{\categ}{\mathpzc} + +\renewcommand{\O}{\mathrm{O}} +\renewcommand{\epsilon}{\varepsilon} +\newcommand{\ph}{\varphi} +\renewcommand{\emptyset}{\varnothing} + +\newcommand{\term}{\textbf} +\newcommand{\rdfn}{=\mathrel{\mathop:}} +\newcommand{\dfn}{\mathrel{\mathop:}=} +\newcommand{\isom}{\simeq} + +\newtheorem{theorem}{Теорема}[subsection] +\newtheorem{lemma}[theorem]{Лемма} +\newtheorem{proposition}[theorem]{Утверждение} + +\theoremstyle{definition} +\newtheorem{example}[theorem]{Пример} +\newtheorem{fact}[theorem]{Факт} + +\newcommand{\la}{\langle} +\newcommand{\ra}{\rangle} +\newcommand{\lAngle}{\langle\!\langle} +\newcommand{\rAngle}{\rangle\!\rangle} +\newcommand{\trleq}{\trianglelefteq} +\newcommand{\ol}{\overline} + +\newcommand{\TBW}{\textbf{TBW}} + +\begin{document} + +\author{Иван Панин\and Виктор Петров} +\title{Мотивы Воеводского и арифметика линейных алгебраических групп +\footnote{Конспект лекций семинара весны 2012 года; предварительная +версия. Автор \TeX-версии~--- Александр Лузгарев. +Основано на конспекте Алексея Бешенова первых двух лекций.}} +\date{2012} +\maketitle + +\section{Введение} + +\subsection{Планы} + +% 13.02.2012 + +Работа Панина и Пименова о квадратичных формах. + +Простая формулировка. {\it Пусть $K = \mathbb{C} (z_1,\ldots,z_n)$ и + $R \dfn \{ \frac{g(z)}{h(z)} \mid h(0) \ne 0 \} \subset K$~--- + регулярные функции в окрестности $0$. Пусть $u \in + R^\times$. Рассмотрим уравнение + +\[ T_1^2 + \cdots + T_k^2 = u. \] + +\noindent (Предполагаем $k \ge 2$.) Если уравнение имеет решение в +$K$, то оно имеет решение и в $R$.} + +\vspace{2em} + +Интересующая нас задача: классифицировать простые алгебраические +группы над произвольным полем (или локальным регулярным кольцом). В +каком смысле~--- мы объясним. Что такое простые алгебраические +группы~--- это обсуждается в записках спецкурса. + +\vspace{2em} + +Как все знают, над алгебраически замкнутыми полями классификацию +простых алгебраических групп дают диаграммы Дынкина. Среди них~--- +четыре бесконечные серии, которым соответствуют следующие +присоединенные группы: + +\begin{itemize} +\item $A_n$~--- $\PGL_{n+1}$. + +\item $B_n$~--- $\SO_{2n+1}$. + +\item $C_n$~--- $\PGSp_{2n}$. + +\item $D_n$~--- $\SO_{2n}$. +\end{itemize} + +Исключительные группы: $E_6$, $E_7$, $E_8$, $F_4$, $G_2$. + +Имеется точная последовательность + +\[ 1 \to \mu_n \to \SL_n \to \PGL_n \to 1. \] + +<<Теорема типа Спрингера>>: \emph{пусть $G$ и $G^\prime$~--- группы типа $G_2$ над полем $K$. Пусть расширение $[L : K]$ нечетное. Тогда если $G_L \isom G_L^\prime$, то $G \isom G^\prime$.} + +С точностью до каких-то тонкостей, имеем + +\[ H^1 (K, G_0^{ad}) \approx \{\text{присоед. простые алг. группы над }K\text{ того же типа, что и }G_0\}. \] + +Это соответствие функториально в том смысле, что расширение полей $L/K$ индуцирует +морфизм $H^1 (K,G^{ad}) \to H^1 (L,G^{ad})$. + +Наша высокая цель~--- построить функтор $F$, сопоставляющий полям +абелевы группы с гомоморфизмом следа, так что конечное расширение +$[L:K]$ давало бы морфизм $F(L) \to F(K)$ и естественное +преобразование + +\[ H^1 (K,G_0^{ad}) \to F(K). \] + +Например, для $G_0 = \PGL_2 = \Aut (M_2)$ ответ такой: + +\[ F\colon K \rightsquigarrow K_2^M (K) / 2, \] + +\noindent где $K_2^M (K) = I^2(K) / I^3 (K)$. + +\subsection{Теорема Меркурьева--Суслина и гипотеза Блоха--Като} + +Пусть $A$~--- центральная простая алгебра над полем $K$ (более общее +понятие~--- \term{алгебра Азумайи}, \term{Azumaya algebra}). Ей +соответствует элемент $[A]$ в группе Брауэра $\Br (K)$. + +\begin{itemize} +\item \textbf{Теорема Меркурьева} (1981)~--- изоморфизм ${}_2 \Br(K) + \isom K_2^M / 2$, а также следствие про $[A] \in {}_2 \Br(K)$. + +[\url{http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0389.0393.ocr.pdf}] + +[\url{http://www.math.ethz.ch/~knus/sridharan/merkurjev84.pdf}] + +\item \textbf{Теорема Меркурьева--Суслина} (1982)~--- изоморфизм ${}_p + \Br(K) \isom K_2^M / p$. + +[L.H. Rowen, Ring theory, Vol. 2, \S 7.2] + +\item \textbf{Гипотеза Блоха--Като} (<>)~--- $K_n^M/p (-) \isom H^n_\text{ét} (-, \mu_p^{\otimes + n})$. + +[\url{http://arxiv.org/abs/0805.4430}] +\end{itemize} + +\subsection{Кольцо Гротендика--Витта} + +$H^1 (K, \O_n)$~--- это классы изометрии невырожденных квадратичных +форм ранга $n$. + +Имеется функтор в кольцо Витта +\[ H^1 (K, \O_n) \to W(K), \quad f \mapsto [f]. \] + +Разберемся, что такое \term{кольцо Витта} $W(K)$. Его образующие~--- +классы изометрии квадратичных форм над $K$, а соотношения выглядят так: +\begin{gather*} +[f] + [g] = [f \perp g],\\ +[f]\cdot [g] = [f\otimes g],\\ +\mathbb{H} = 0, +\end{gather*} +где $\mathbb{H}$~--- класс изометрии двумерной квадратичной формы $f(x,y)=xy$, а +$f \perp g$ имеет следующий смысл. Если $f$~--- квадратичная форма на +$V$, а $g$~--- квадратичная форма на $W$, то на $V\oplus W$ задается +квадратичная форма $(f\perp g) (u\oplus v) \dfn f(u) + g(v)$. + +Имеется корректно определенный гомоморфизм +\begin{eqnarray*} +\rk\colon W(K) & \to & \mathbb{Z}/2,\\ +{}[f] & \mapsto & \rk f \mod 2. +\end{eqnarray*} +$I \dfn \ker \rk$ называется \term{фундаментальным идеалом}. + +\term{Кольцо Гротендика--Витта} $GW (K)$ определяется следующим +образом. В нем те же образующие, но нет условия $[xy] = 0$. Сначала +определяется сложение +и умножение, делающее $GW (K)$ полукольцом: +\begin{gather*} +[f] + [g] = [f \perp g],\\ +[f]\cdot [g] = [f\otimes g]. +\end{gather*} +Потом мы берем группу Гротендика и получаем кольцо. + +\subsection{$\mathbb{A}^1$-гомотопии и гипотеза Мореля} + +[\url{http://mathunion.org/ICM/ICM1998.1/Main/00/Voevodsky.MAN.ocr.pdf}] + +$\mathbb{A}^1$-гомотопическая категория пространств с отмеченными +точками над $K$. + +Сфера $S^0 = \{ +, \bullet \}$ состоит из двух точек, из которых +$\bullet$~--- отмеченная. + +Теорема Мореля (1999?) состоит в вычислении + +\[ \pi_0^{stab} (S^0) \isom \GW (K) \] + +Fabien Morel, On The Motivic $\pi_0$ of the Sphere Spectrum.\\ +\url{http://dx.doi.org/10.1007/978-94-007-0948-5_7} + +Желаемый функтор $F$ мог бы давать $H^1 (K,G) \to H_0^{\mathbb{A}^1} (B^\text{èt} G)$. + +Аналог этого в топологии следующий. Пусть задано главное +$G$-расслоение $\mathfrak{g} \to X$ для клеточного пространства +$X$. Сопоставим ему отображение в классифицирующее пространство $X +\xrightarrow{f_\mathfrak{g}} BG$. + +Существует соответствие между множеством классов изоморфности главных +$G$-расслоений над $X$ и гомотопическими классами $[X,BG]$. + +Имеется инъекция + +\[ [X,BG] = \pi_0 (\Map (X,BG)) \hookrightarrow H_0 (\Map (X,BG)). \] + +Гипотеза Мореля заключается в том, что в алгебраической ситуации тоже получается инъекция + +\[ [\Spec K, B^\text{èt} G] = \pi_0^{\mathbb{A}^1} (B^\text{èt} G) +\hookrightarrow H_0^{\mathbb{A}^1} (B^\text{èt} G). \] + +\subsection{Формы Пфистера} + +Рассмотрим фильтрацию на кольце Витта + +\[ W(K) \supset I \supset I^2 \supset I^3 \supset \cdots \] + +\begin{theorem} +$\bigcap_n I^n = \{0\}$. +\end{theorem} + +Мы уже знаем, что $W(K) / I = \mathbb{Z}/2$. +$I/I^2$ как абелева группа порождается элементами вида $\lAngle a\rAngle \dfn x^2 - a\,y^2$ для некоторого $a\in K^\times$. +Более общо, $I^n/I^{n+1}$ порождается тензорными произведениями элементов +\[ \lAngle a_1, \ldots, a_n\rAngle = \lAngle a_1\rAngle\otimes\cdots\otimes\lAngle a_n\rAngle. \] +$\lAngle a_1, \ldots, a_n\rAngle$ называется \term{$n$-кратной формой Пфистера}. + +\begin{example} +При $n = 1$ имеем $a \in K^*/(K^*)^2$; $K (\sqrt{a})$~--- квадратичное расширение. + +При $n = 2$ символ $\lAngle a,b\rAngle$ есть норма алгебры кватернионов +$H = (a,b)$ над $K$. + +При $n = 3$ символ $\lAngle a,b,c\rAngle$ есть норма алгебры октонионов +$(a,b,c)$ над $K$ (что соответствует группам типа $G_2$ над $K$). +\end{example} + +\begin{theorem}[Арасон] +Если $[q] \in I^n$, то $\rk q \ge 2^n$. +Если при этом $\rk q = 2^n$, то $q \isom \alpha \cdot \lAngle +a_1,\ldots,a_n \rAngle$, где $\alpha \in K^\times$. +В частности, $\bigcap I^n = 0$. +\end{theorem} + +\begin{itemize} +\item $e_0 (q) \dfn \rk q \mod 2$. + +\item Если $e_0 = 0$, то $[q] \in I$. Определим $e_1 (q) \dfn [\![q]\!] \in I/I^2$. Этому соответствует $\lAngle a\rAngle$, где $a$~--- дискриминант $q$ (с точностью до знака?). + +\item Если $e_1 = 0$, то $e_2 (q) \dfn [\![ q ]\!] \in I^2/I^3$. +Форме $q$ можно сопоставить $C_0^+ (q)$, четную положительную часть алгебры +Клиффорда, это будет центральная простая алгебра. Имеем $[C_0^+ (q)] +\in {}_2 \Br (K)$. По теореме Меркурьева, это сумма +\[ [(a_1,b_1)]\,[(a_2,b_2)]\cdots [(a_k,b_k)] \] + +\[ \lAngle a_1,b_1\rAngle + \lAngle a_2,b_2\rAngle + \cdots +\lAngle a_k,b_k\rAngle. \] + +\item Если $e_2 (q) = 0$, то можно определить $e_3 (q)$~--- + \term{инвариант Арасона}. +\end{itemize} + +\subsection{Торсоры} + +Пусть $G$~--- простая алгебраическая группа над $K$. + +\term{$G$-торсором} называется многообразие $X$ над $K$, такое что +\begin{itemize} +\item определено действие $G\times X\to X$; + +\item над алгебраическим замыканием $\overline{K}$ имеется изоморфизм + $X_{\overline{K}} \isom G_{\overline{K}}$ (как многообразий с + $G$-действием). +\end{itemize} + +Раньше торсоры назывались <<главными однородными пространствами>> +(principal homogeneous space). + +\begin{example} +Действие $G$ сдвигами на себе дает \term{тривиальный $G$-торсор}. +\end{example} + +По определению, $H^1 (K;G)$ есть множество классов изоморфности +$G$-торсоров с отмеченной точкой (тривиальный $G$-торсор). + +\begin{example} +Зафиксируем $a\in K$. +Для каждой $K$-алгебры $R$ положим +$\mu_2(R) = \{ x\in R\mid x^2 = 1 \}$, $X(R) \dfn \{ y\in R\mid y^2 = a\}$. +Получаем схемы $\mu_2$ и $X$, причем +$\mu_2$ действует на $X$ умножением: +если $y^2 = a$, $x^2 = 1$, то $(x\,y)^2 = a$. +\end{example} + +$X$~--- тривиальный $G$-торсор iff у него есть рациональная точка: $X +(K) \ne \emptyset$. + +Если $G$~--- абелева группа, то на торсорах имеется сложение. При этом +$H^1 (K,\mu_2) \isom K^* / (K^*)^2$ как абелева группа. И вообще $H^1 +(K,\mu_n) \isom K^* / (K^*)^n$. + +\subsection{Скрученные формы} + +Напомним, что $\O_{2n} = \Aut (q_{split})$, где $q_{split} = x_1\,y_1 + \cdots + +x_n\,y_n$~--- \term{расщепимая форма} (от переменных $x_1,\dots,x_n,y_1,\dots,y_n$. + +$H^1 (K, \O_{2n})$ можно отождествить с множеством классов изометрии +невырожденных квадратичных форм ранга $2n$. +Действительно, пусть $q$~--- квадратичная форма, +Мы утверждаем, что $\Iso (q_{split}, +q)$ есть искомый торсор: на нем действует $\O_{2n}$. +Здесь $\Iso(\ph,\psi)$ обозначает функтор изоморфизмов между квадратичными +формами $\ph$ и $\psi$; более точно, +$\Iso(\ph,\psi)(R) = \{f\colon\ph_R\to\psi_R\mid\mbox{$f$~--- изоморфизм}\}$. +Над алгебраически +замкнутым полем $q$ изоморфна $q_{split}$, и получается $\Iso +(q_{split},q_{split})=\Aut (q_{split})=\O_{2n}$. + +Пусть $A$~--- некоторая алгебраическая структура над полем $K$ +(например, квадратичное пространство, конечномерная ассоциативная +алгебра, конечномерная неассоциативная алгебра). Тогда +\term{скрученная форма $A^\prime$} для $A$ есть такая структура над +$K$, что при переходе к алгебраическому замыканию +$A^\prime_{\overline{K}} \isom A_{\overline{K}}$. + +\noindent\textbf{Теорема}. $H^1 (K, \Aut(A))$ есть множество классов +изоморфности скрученных форм $A$ над $K$. + +Изоморфизм такой: +\[ A' \xmapsto{\sim} \Iso (A,A^\prime). \] +На $\Iso (A,A^\prime)$ есть структура алгебраического многообразия. + +\vspace{2em} + +\noindent\textbf{Замечание}. Пусть $X$~--- проективное многообразие +над $K$. Теорема (Гротендик): \emph{функтор $U \mapsto \Aut_U (X\times + U)$ представим в схемах}; то есть, существует схема $R$ такая, что +$\Aut_U(X\times U)$ естественно изоморфно $\Hom(U,R)$. + +\vspace{2em} + +Контрпример: $\Aut (\mathbb{A}^n)$ не конечномерно. + +Пример: $A \dfn M_n (K)$. $\Aut (A) = \PGL_n$.\\ +$H^1 (K, M_n(K))$~--- это скрученные формы $M_n (K)$, то есть +центральные простые алгебры размерности $n^2$, взятые с точностью до +изоморфизма. + +$\Aut (\mathbb{P}^{n-1}) = \PGL_n = \GL_n / \mathbb{G}_m$. + +Автоморфизмы сохраняют ранг. + +$H^1 (K, \PGL_n)$ есть множество скрученных форм $\mathbb{P}^{n-1}$ +над $K$ = \term{многообразия Севери--Брауэра}. + +\[ A \mapsto \SB (A) = \{ \text{левые идеалы $I\trleq A$}\mid +\dim_K(I)=n \} \] + +Пример при $n=2$: кватернионы $A = (a,b)$. + +$\beta\,u + \gamma\,v + \delta\,u\,v$. Имеем векторное пространство +$u,v,uv$. Условие $\{ \text{норма} = 0 \}$ задает проективное +подмногообразие в $\mathbb{P}^2$. + +$x^2 - a\,y^2 - b\,z^2 = 0$~--- коника. + +\begin{eqnarray*} +\PGL_2 & \isom & \SO_3, \\ +\{\text{кватернионы}\} & \isom & \{\text{формы ранга }3\text{ с + трив. дискриминантом}\}. +\end{eqnarray*} + +\subsection{Точные последовательности алгебраических групп} + +Точность последовательности алгебраических групп над $K$ + +\[ 1 \to C \to H \to G \to 1 \] + +\noindent означает следующее: + +\begin{enumerate} +\item $C$~--- алгебраическая подгруппа в $H$. + +\item После расширения скаляров $H (\overline{K})\to G (\overline{K})$ + является сюръекцией + \emph{над алгебраическим замыканием поля $K$}. + +\item $C = \ker (H\to G)$, $C(R) = \ker (H(R) \to G(R))$. +\end{enumerate} + +\begin{example} +Следующая последовательность алгебраических групп точна в указанном смысле: + +\[ 1 \to \mu_2 \to \mathbb{G}_m \to \mathbb{G}_m \to 1, \] + +\noindent где $\mathbb{G}_m (K) \dfn \{ (x,y)\in K^2 \mid x\,y = 1 \}$, и +отображение $\mathbb{G}_m \to \mathbb{G}_m$ есть $x \mapsto x^2$ (это +сюръекция над алгебраическим замыканием). +\end{example} + +\begin{example} +Следующая последовательность точна: + +\[ \mu_2 (K) \to K^\times \to K^\times \to H^1 (K,\mu_2) \to H^1 +(K,\mathbb{G}_m) \to H^1 (K,\mathbb{G}_m). \] +\end{example} + +\noindent (Отображение $K^\times \to K^\times$ есть $x \mapsto x^2$.) + +\begin{theorem}[Теорема Гильберта 90] + +\begin{gather*} +H^1 (K,\mathbb{G}_m) = \{\bullet\},\\ +H^1 (K,\GL_n) = \{\bullet\}. +\end{gather*} +\end{theorem} + +Из точности последовательности выше и теоремы 90 получается + +\[ H^1 (K,\mu_2) \isom K^\times / (K^\times)^2. \] + +\begin{example} +Точная последовательность +\[ 1 \to \SL_n \to \GL_n \xrightarrow{\det} \mathbb{G}_m \to 1. \] +приводит к последовательности +\[ \GL_n (K) \xrightarrow{\det} \mathbb{G}_m (K) \to H^1 (K,\SL_n) \to H^1 (K,\GL_n). \] + +Здесь $H^1 (K,\SL_n) = \{\bullet\}$ и $H^1 (K,\GL_n) = \{\bullet\}$. +\end{example} + +\begin{example} +\[ 1 \to \mu_n \to \SL_n \to \PGL_n \to 1. \] + +\[ \mu_n (K) \to \SL_n (K) \to \PGL_n (K) \to K^\times / (K^\times)^2 \to \{\bullet\} \to H^1 (K,\PGL_n). \] + +\[ 1 \to \mathbb{G}_m \to \GL_n \to \PGL_n \to 1. \] +\end{example} + +\begin{example} +\begin{eqnarray*} +\SL_n & \to & \PGL_n,\\ +g & \mapsto & (x \mapsto g\,x\,g^{-1}) \in \Aut (M_n). +\end{eqnarray*} + +Это сюръекция алгебраических групп, но не сюръекция на точках. + +\[ 1 \to \mu_n \to \SL_n \to \PGL_n \to 1. \] +\end{example} + +\begin{theorem} +Если имеется точная последовательность $1 \to C \to H \to G \to 1$, то +возникает точная последовательность множеств с отмеченной точкой + +\[ 1 \to C(K) \to H(K) \to G(K) \to H^1 (K,C) \to H^1 (K,H) \to H^1 (K,G). \] +\end{theorem} + +См. книгу Серра <<Когомологии Галуа>>. + +% 27.02.2012 + +\subsection{Вторые когомологии} + +\begin{itemize} +\item Напомним, что $H^1 (F,G)$~--- множество $G$-торсоров. \emph{Если $G$ + коммутативна}, то это аффинная алгебраическая группа. + +(Как в этом случае умножаются торсоры?~--- Что-то типа $E_1 \mathop{*} +E_2 = (E_1 \times E_2) / ((e_1,e_2) = (g\,e_1,g\,e_2))$.) + +\item $H^0 (F,G)$~--- это функтор $F \rightsquigarrow G(F)$, + т.е. функтор точек. \emph{Если $G$ коммутативна}, то $H^i (F,G)$ + можно определить как $i$-й производный функтор. При $i = 1$ это + совпадает с первым определением. +\end{itemize} + +\begin{theorem} +Пусть имеется точная последовательность $1 \to C \to G \to H \to +1$. Предположим, что $C \le \Cent (G)$. Тогда точная +последовательность продолжается до вторых когомологий: + +\[ 1 \to C(F) \to G(F) \to H(F) \to H^1 (F,C) \to H^1 (F,G) \to H^1 (F,H) \to H^2 (F,C). \] +\end{theorem} + +\begin{example} +$H^2 (F, \mathbb{G}_m)=\Br (F)$~--- \emph{группа Брауэра} поля $F$: +она состоит из классов эквивалентности $[A]$ центральных простых +алгебр $A$ над $F$; умножение выглядит так: $[A]\cdot +[B]=[A\otimes_FB]$. + +Пусть $X$~--- квазипроективное многообразие. Тогда $H^2 +(X,\mathbb{G}_m)_{tors} = \Br (X)$ (\emph{теорема Габбера} (Gabber)). +(Загадочное замечание: +подразумевается топология fppf, а для этальной топологии в определении +торсора вместо $\overline{F}$ нужно взять $F^{sep}$.) +\end{example} + +\begin{example}[Топологический аналог] +Пусть $X$~--- хорошее топологическое пространство (например, область в +$\mathbb R^n$, многообразие или CW-комплекс). + +Пусть $G$~--- топологическая группа (например, $S^1$, $S^3$, $\SL_2 +(\mathbb{C})$, $\O_n (\mathbb{C})$). + +Имеется левое действие $G \times (G\times X) \to (G\times X)$, $g_1 +\cdot (g_2,x) \mapsto (g_1\,g_2, x)$. + +Левое действие послойно и свободно на скрученной форме $G\times +\mathcal{G} \to \mathcal{G}$. + +Для всех $x \in X$ возникает действие $G \times \mathcal{G} (x) \to +\mathcal{G} (x)$. Здесь $\mathcal{G} (x) \isom G$, и этот изоморфизм +зависит от $x$. + +$(\mathcal{G}, G\times \mathcal{G} \to \mathcal{G})$ в топологии +называется \term{главным $G$-расслоением} (\term{principal + $G$-bundle}). + +$\mathcal{G}/G = X$. +\end{example} + +\begin{example} +$\mathbb{C}^\times = \GL_1 (\mathbb{C}) = \Aut (\mathbb{C}^1)$. + +Пусть $L \to X$~--- комплексное линейное расслоение, $z (X)$~--- +нулевое сечение. + +Рассмотрим отображение $\mathbb{C}^\times \times (L - z(X))\to (L - +z(X))$, $(\lambda, v)\mapsto \lambda v$. Имеем изоморфизм $L(x) - +0\isom \mathbb{C}^\times$, зависящий +от $x$. + +\begin{itemize} +\item Тогда $H^1 (X,\mathbb{C}^\times)$~--- классы изоморфизма + $\mathbb{C}^\times$-торсоров над $X$. Они соответствуют линейным + расслоениям над $X$: расслоению $L$ соответствует описанный выше + торсор $L - z(X)$, и по торсору $\mathcal G^\times$ можно построить + расслоение $\mathcal L$. + +\item Таким образом, мы видим, что $H^1 (X, \Aut (\mathbb{C}^1))$~--- + это скрученные формы расслоения + $\mathbb{C}\times X$ над $X$. + +\item Аналогично, $H^1 (X, \Aut (\mathbb{C}^n))$~--- это (1) скрученные формы + расслоения $\mathbb{C}^n\times X$ над $X$, то есть (2) векторные + расслоения над $X$ со слоем $\mathbb{C}^n$ (с точностью до изоморфизма). + +\item Пусть $\Aut_{\mathbb{C}} (\mathbb{C}^{2n}, \sum u_i\,v_i)$~--- + автоморфизмы, сохраняющие квадратичную форму. + Тогда \[H^1 (X, \Aut_{\mathbb{C}} (\mathbb{C}^{2n}, \sum + u_i\,v_i))\]--- это (1) скрученные формы расслоений вида + $(\mathbb{C}^n\times X, \sum u_i,v_i) \to X$, то есть (2) векторные + расслоения $E \to X$ со слоем $\mathbb{C}^n$ и с квадратичной формой в + слоях. + +\item Рассмотрим $\Aut (M_n (\mathbb{C})) = \PGL_n + (\mathbb{C})$. Тогда $H^1 (X, \Aut (M_n (\mathbb{C})))$~--- это + скрученные формы расслоений вида $M_n (\mathbb{C})\times X \to + X$. Например, по каждому расслоению $E\to X$ можно построить + расслоение $\End(E)\to X$, и послойно $\End(E)(x)=\End(E(x))$. Но + бывают и расслоения, не изоморфные никакому $\End(E)\to X$~--- это + нетривиальные топологические алгебры Адзумайи. +\end{itemize} +\end{example} + +Имеется точная последовательность + +\[ 1 \to \mathbb{C}^\times \to \GL_n (\mathbb{C}) \to \PGL_n (\mathbb{C}) \to 1. \] + +Отсюда получается точная последовательность + +\begin{gather*} +1 \to \Gamma (X, \mathbb{C}^\times) \to \Gamma (X, \GL_n (\mathbb{C})) +\to \Gamma (X, \PGL_n (\mathbb{C})) \to \\ \to H^1 (X,\mathbb{C}^1) +\to H^1 (X,\GL_n (\mathbb{C})) \to H^1 (X, \PGL_n (\mathbb{C})) \to +H^2 (X, \mathbb{C}^\times). +\end{gather*} + +\term{Топологическая группа Брауэра} есть $\Br_{top} (X) \dfn +H^2_{top} (X, \mathbb{C}^\times)$. + +\begin{example} +Мы утверждаем, что +\[ H^2 (X, S^1) \twoheadrightarrow H^3 (X, \mathbb{Z})_{tors}. \] + +Заметим, что +$\mathbb{C}^\times \isom S^1 \times \mathbb{R}$. Имеем точную +последовательность +\[ 0 \to \mathbb{Z} \to \mathbb{R} \to S^1 \to 0, \] +откуда получаем точную последовательность +\[ H^2(X,\mathbb{Z})\to H^2(X,\mathbb{R}\to H^2(X,S^1)\to +H^3(X,\mathbb Z)\to H^3(X,\mathbb R)=H^3(X,\mathbb Z)\otimes \mathbb R.\] +Обозначим отображение $H^3(X,\mathbb Z)\to H^3(X,\mathbb +Z)\otimes\mathbb R$ через $\alpha$. Тогда связывающий гомоморфизм +дает нам отображение $H^2(X,S^1)\to\ker(\alpha)=H^3(X,\mathbb Z)_{tors}$. + +На самом деле, +\[ \Br_{top} (X) = H^3 (X,\mathbb{Z})_{tors}. \] +Нечто такое написано как определение (у Серра? Гротендика?). +\end{example} + +А какие нам известны нетривиальные скрученные формы алгебры $M_n (K)$? +Так это и есть центральные простые алгебры. + +Имеем точную последовательность $1 \to \mathbb{G}_m \to \GL_n \to +\PGL_n \to 1$, откуда + +\[ H^1 (F,\GL)_n \to H^1 (F, \PGL_n) \to H^2 (F,\mathbb{G}_m). \] + +При этом +$H^1 (F,\GL)_n = \{ \bullet \}$, и $H^1 (F, \PGL_n)$~--- центральные +простые алгебры степени $n$. Это отображение дает изоморфизм % ??? + +\begin{eqnarray*} +\Br (F) & \isom & H^2 (F,\mathbb{G}_m),\\ +A & \mapsto & [A]. +\end{eqnarray*} +% тут пропущен кусок про умножение в H^1??? + +Имеется точная последовательность +\[ 1 \to \mu_n \to \mathbb{G}_m \to \mathbb{G}_m \to 1, \] +где $\mathbb{G}_m \to \mathbb{G}_m$~--- отображение $x \mapsto x^n$. + +Получаем точную последовательность +\[ \xymatrix{ H^1 (F,\mathbb{G}_m) \ar[r]\ar@{=}[d] & H^2 (F,\mu_n) + \ar[r] & H^2 (F,\mathbb{G}_m) \ar[r]\ar@{=}[d] & H^2 + (F,\mathbb{G}_m)\ar@{=}[d] \\ +\{ \bullet \} & & \Br (F) \ar[r]^{\cdot n} & \Br (F) }, \] +откуда +$H^2 (F,\mu_n) = {}_n \Br (F)$. + +Еще один пример: пусть $\fchar F \ne 2$. Имеется точная последовательность +\[ 1 \to \SO_n \to \O_n \xrightarrow{\det} \mu_2 \to 1. \] + +Тогда +\[ \O_n (F) \twoheadrightarrow \mu_2 (F) \to H^1 (F,\SO_n) \to H^1 (F,\O_n) \xrightarrow{\disc} H^1 (F,\mu_2)=F^*/(F^*)^2. \] + +Отсюда $H^1 (F,\SO_n)$ (невырожденные квадратичные формы дискриминанта +1)~--- подмножество в $H^1 (F,\O_n)$ (невырожденные квадратичные формы +ранга $n$ с точностью до изометрии). + +Это дает нам инвариант $\disc=e_1\colon I\to I/I^2\isom F^*/(F^*)^2$. + +Еще один пример: +\[ 1 \to \mu_2 \to \Spin_n \to \SO_n \to 1. \] + +\[ \Spin_n (F) \to \SO_n (F) \xrightarrow{N} H^1 (F,\mu_2) \xrightarrow{0} H^1 (F,\Spin_n) \to H^1 (F,\SO_n) \xrightarrow{e_2} H^2 (F,\mu_2). \] + +Здесь +\begin{itemize} +\item $H^2 (F,\mu_2) = {}_2 \Br (F)$. + +\item $N$~--- \term{спинорная норма}. А именно, каждый элемент $\SO_n$ + раскладывается в произведение отражений $g = S_{v_1} \cdots + S_{v_{2k}}$; тогда $N(g) \dfn q (v_1) \cdots q (q_{2k}) + \pmod{(F^\times)^2}$. + +\item Отображение $\Spin_n(F)\to\SO_n(F)$ уже не обязательно является + сюръективным. + +\item Мы не знаем, что такое $H^1(F,\Spin_n)$. Отображение + $H^1(F,\mu_2)\to H^1(F,\Spin_n)$ равно $0$ по теореме Эйхлера. +\end{itemize} + +Самая правая стрелка в этой длинной последовательности дает нам инвариант +$e_2\colon I^2 \to I^2 / I^3 = {}_2 \Br (F)$. Его можно описать так: +по форме $q$ можно построить алгебру Клиффорда $C(q)$ с четной частью +$C_0(q)$. Тогда $e_2$ сопоставляет форме $q\in I^2$ класс $[C^+_0(q)]$ +в $\Br(F)$. + +Пусть $E$~--- левый $G$-торсор, $G$ действует на $X$ справа. +Рассмотрим скрученную форму $X$ + +\[ {}_EX \dfn (X\times E)/ \! {(x,e) \sim (x\,g^{-1}, g\,e)}. \] + +На ней действует ${}_EG$. Действительно, +${}E_G=\Aut_{G-\text{торс}}(E)$~--- автоморфизмы $E$ как +$G$-торсора. ${}_EG$ является группой (тут $G$ действует сопряжениями +на себе). + +% тут пропущен кусок??? + +\begin{example} +Рассмотрим $H^1 (F,\O_n)$. + +$E \in H^1 (F,\O_n)$ задается квадратичной формой $q$, и $q$ должна +быть формой расщепимой квадратичной формы $q_0$. При этом $E = \Isom +(q_0, q)$. + +$O (q_0)$ действует на квадрике $Q_0 \dfn \{ q_0 = x_1\,y_1 + \cdots + +x_n\,y_n = 0 \} \subseteq \mathbb{P}^{2n-1}$. + +После подкрутки: ${}_EQ_0 = \{ q = 0 \}$ и на ${}_EQ_0$ действует +группа $O(q)={}_EO(q_0)$. +\end{example} + +\subsection{Многообразия Севери--Брауэра} + +\begin{example} +Рассмотрим $H^1 (F, \PGL_n)$. + +Торсор $E\in h^1(F,\PGL_n)$ задается центральной простой алгеброй $A$ +степени $n$: +$E = \Isom_{F\text{-}\categ{Alg}} (M_n, A)$. Напомним, что $\PGL_n=\Aut(M_n)$. + +%Что такое $\mathbb{P}^{n-1}$? + +Каждому вектору $v \in \mathbb{A}^n - \{ 0 \}$ соответствует правый +идеал $\{x\mid \im x\leq \left\}$ в $M_n$. Множество всех +идеалов, получающихся таким образом~--- это в точности множество +правых идеалов размерности $n$. + +%$\left \rightsquigarrow \text{правый идеал в } M_n \{ x \mid +%\im x \subseteq \left \}$. + +${}_E\mathbb{P}^{n-1}$~--- Множество правых идеалов размерности $n$ в +$A$~--- \term{многообразие Севери--Брауэра} $\SB (A)$. + +Уравнения: +\[ \SB (A) \dfn \{ W \subset A \mid W\cdot A \subseteq A \}. \] + +Таким образом, +\[ \SB (A) \hookrightarrow \Gr (n,A) = \Gr (n,n^2). \] + +\[ \xymatrix{ +A\times \SB(A) & A\times \Gr (n,n^2) \\ +\left.\tau\right|_{\SB (A)}\ar[d]\ar@{^(->}[u] & \tau_n\ar[d]\ar@{^(->}[u] & W\ar[d] \\ +\SB (A)\ar@{^(->}[r] & \Gr (n,n^2) & \{ w \} +} \] +\end{example} + +\begin{lemma} +$\End_{\SB (A)} (J_A) \isom A$ (эндоморфизмы расслоения). +\end{lemma} +Поэтому два описания $H^1(F,\PGL_n)$~--- как алгебры Адзумайи и как +формы $\mathbb P^{n-1}$~--- эквивалентны. + +\begin{proposition} +Подрасслоение $\left.\tau\right|_{\SB(A)}$ выдерживает правое +$A$-действие на $A\times \SB(A)$. + +$J_A \dfn \left.\tau\right|_{\SB (A)}$. +\end{proposition} + +\begin{example} +$\Gr (K,n)$ (линейные $k$-мерные подпространства в $\mathbb{A}^n$). + +Если $U$~--- $k$-мерное подпространство в $\mathbb{A}^n$, то +$\{ x \mid \im x \subseteq U \}$~--- правый идеал в $M_n$ размерности +$kn$. + +$\SB_k (A)$~--- обобщенное многообразие Севери--Брауэра~--- многообразие +правых идеалов размерности $k$. +\end{example} + +$\Gr (k,n) \isom \Gr (n-k, n)$ (напомним, что это не канонический +изоморфизм). Аналог этой двойственности: $\SB_k (A) \isom \SB_{n-k} +(A^{op})$. + +\begin{proposition} +$\SB (A) (F) \ne \emptyset \Rightarrow A \isom M_n$. + +$\SB_k (A) (F) \ne \emptyset \Rightarrow \ind A \mid k$. +\end{proposition} + +(Напомним, что такое $\ind$. Для центральной простой алгебры $A$ имеем +$A \isom M_m (D)$, где $D$~--- тело. $m \cdot \deg D = n$. $\deg D \rdfn +\ind A$, где $\deg D \dfn \sqrt{\dim D}$.) + +Скрученные формы $\mathbb{P}^{n-1}$~--- это скрученные формы +$M_n$. Имеем $\Aut (\mathbb{P}^{n-1}) = \PGL_n$. + +Предположим $\fchar F \ne 2$. + +$\Aut(q_0) = O(q_0)$. + +$\Aut (Q_0)^+ = \PGO (q_0)$, где $Q_0$~--- квадрика $\{ q = 0 \}$. + +$H^1 (F, \PGO (q_0))$~--- классы $(A,\sigma)$ изоморфности центральных +простых алгебр $A$ с ортогональной инволюцией $\sigma$. + + +\[ \{ \text{правые идеалы }I\text{ в }(A,\sigma)\text{ размерности }\deg A \mid \sigma (I) \cdot I = 0 \} \rdfn X_{(A,\sigma)} \hookrightarrow \SB(A). \] + +При поднятии до $\overline{F}$ получаем: +\[ Q_{\sigma (q_{\overline{F}})} \hookrightarrow \mathbb{P}^{\deg A - + 1}_{\overline{F}} = \SB (A) \otimes_F \overline{F}. \] + +Вложение $X_{(A,\sigma)} \hookrightarrow \SB(A)$ есть аналог вложения +квадрики в проективное пространство. + +% 05.03.2012 + +\section{Проективные однородные многообразия} + +\subsection{Первые примеры} + +Еще раз про аналогию с топологией: + +$E\to X$~--- торсор на топологическом пространстве $X$, $G$ действует +на $E$. Существует покрытие $\{U_i\}$ пространства $X$ такое, что +\[\begin{xymatrix}{U_i\times G\isom E|_{U_i}\ar[r]\ar[d] & E \\ +U_i\ar@{^(->}[r] & X}\end{xymatrix}\] + +У нас: возьмем $X=\Spec K$. Пусть $E\to\Spec K$~--- торсор. Существует +расширение полей $L/K$ такое, что торсор $E_L\to\Spec L$ изоморфен +торсору $G_L\to\Spec L$. + +Мы хотим описать $H^1(K,G)$. Стратегия: для торсора $E$ и (гладкого +проективного) $G$-многообразия $X$ мы определили ${}_EX$~--- +${}_EG$-многообразие (снова гладкое проективное), которое называется +\emph{скрученной формой $X$}, то есть, +\begin{itemize} +\item $E_{\overline{K}}\isom +G_{\overline{K}}$ как $G_{\overline{K}}$-многообразие, +\item $({}_EG)_{\overline{K}}\isom G_{\overline{K}}$ как алгебраическая +группа, +\item $({}_EX)_{\overline{K}}\isom X_{\overline{K}}$ как + $G_{\overline{K}}$-многообразие. +\end{itemize} + +\begin{example} +Пусть $A\in H^1(K,\PGL_n)$; то есть, $A$~--- центральная простая +алгебра степени $n$. Положим $E=\Isom(M_n,A)$, $X=\mathbb P^{n-1}$, +$G=\PGL_n$. Тогда ${}_EG=\Aut(A)$, ${}_EX=\SB(A)$~--- многообразие +правых идеалов в $A$ размерности $n$. Заметим, что $\SB(A)(K)$ непусто +тогда и только тогда, когда $A\isom M_n$. Вообще, свойства многообразия +$\SB(A)$ отражают свойства исходного торсора. +\end{example} + +\begin{example} +Пусть $G=\O_n$, $q\in H_1(K,\O_n)$~--- невырожденная квадратичная +форма ранга $n$. + +$E=\Isom(q_0,q)$, где $q_0$ расщепима (то есть, имеет +вид $\langle 1,-1\rangle\perp\dots\perp\langle 1,-1\rangle$ плюс, +возможно, слагаемое $\langle 1\rangle$). + +$X=\{q_0=0\}$ в проективном смысле. Тогда $Q={}_EX=\{q=0\}$. Заметим, +что $Q(K)$ непусто тогда и только тогда, когда форма $q$ изотропна, то +есть, $q\isom\langle 1,-1\rangle\perp q'$. Этот факт остается верным +при любом расширении $L/K$: $Q(L)$ непусто тогда и только тогда, когда +форма $q_L$ изотропна. +\end{example} + +\begin{fact} +Пусть $q$ имеет вид $\lAngle a_1,\dots,a_k\rAngle=\la +1,-a_1\ra\otimes\dots\otimes\la 1,-a_k\ra$. Тогда +\begin{multline} +\text{ +Форма $q_L$ изотропна тогда и только тогда, когда она расщепима (то +есть,} \\ +\text{раскладывается в сумму форм вида $\la 1,-1\ra$).}\tag{*} +\end{multline} +Наоборот, если $\dim q$ четна и (*) выполнено для любого расширения +полей, то $q$ пфистерова с точностью до скаляра. Если же $\dim q$ +нечетна, то $q\perp\la 1\ra$ пфистерова с точностью до скаляра. +\end{fact} + +Таким образом, от торсора $E$ можно переходить к многообразию ${}_EX$ +(и можно варьировать $X$), смотреть на его инварианты (в смысле +алгебраической геометрии) и получать отсюда информацию об инвариантах +торсора. + +Пусть $X$~--- гладкое проективное многообразие. Мы ограничимся +случаем, когда $X$ \emph{однородное}, то есть, $G(\overline{K})$ +действует на $X(\overline{K})$ транзитивно (заметим, что это означает, +что отображение $G\times X\to X\times X$, $(g,x)\mapsto (gx,x)$ +сюръективно как пучок, а не в категорном смысле; категорное понятие +эпиморфизма не подходит для наших целей: например, отображение +$\Spec{\mathbb Q}\to\Spec{\mathbb Z}$ сюръективно в категории схем). +Неформально говоря, у $G$ на $X$ одна орбита. +Тогда $X$ называется \term{проективным однородным многообразием}. + +Как строить проективные однородные многообразия? Пусть $G$~--- +расщепимая группа, $V$~--- неприводимое представление (в положительной +характеристике нужно действовать осторожене). Рассмотрим $\mathbb +P(V)$~--- многообразие прямых в $V$, проходящих через $0$. У группы +$G$ есть ровно одна замкнутая орбита на $\mathbb P(V)$~--- это и есть +наше $X$. На самом деле, все проективные однородные многообразия так +получаются (но не обязательно единственным образом). + +\begin{example} +$G=\SL_n$ действует на $V=K^n$ (имеется в виду обычное, \emph{векторное} +представление). Пусть $u$, $v$~--- два вектора. Можно ли найти $g$ +такое, что $\la gu\ra=\la v\ra$ (здесь через $\la +x\ra$ мы обозначаем прямую, натянутую на $x$)? Ответ~--- можно, +если $u$ и $v$ отличны от $0$. Значит, в $K^n$ есть две орбиты +действия группы $G$: $\{0\}$ и $\{v\mid v\neq 0\}$. После +проективизации в $\mathbb P(K^n)=\mathbb P^{n-1}$ остается только одна +орбита. +\end{example} + +\begin{example} +$G=\SL_n$ действует на $V=\Lambda^k(K^n)$, $k=1,\dots,n-1$. На +неразложимых поливекторых орбит много, но на разложимых действие +транзитивно. Свойство <<быть разложимым>> определяется уравнениями +Плюккера. Орбита в $\mathbb P(\Lambda^k(K^N))$~--- это $\Gr(k,n)$. + +Пусть, к примеру, $n=4$, $k=2$. Диаграмма Хассе весов нашего +представления выглядит так: +\[ +\begin{tikzpicture}[scale=1.0,thin, +aln/.style={above left=-2pt}, +arn/.style={above right=-2pt}, +bln/.style={below left=-2pt}, +brn/.style={below right=-2pt}, +every label/.style={above=2pt}] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.4, 1)$); +\coordinate (4) at ($\sm*(2.4, -1)$); +\coordinate (5) at ($\sm*(3.4, 0)$); +\coordinate (6) at ($\sm*(4.8, 0)$); +\draw (1)--node[above] {$2$}(2); +\draw (2)--node[aln] {$1$}(3); +\draw (2)--node[bln] {$3$}(4); +\draw (3)--node[arn] {$3$}(5); +\draw (4)--node[brn] {$1$}(5); +\draw (5)--node[above] {$2$}(6); +\foreach \point in +{1,2,3,4,5,6} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\end{tikzpicture} +\] +Разложимый тензор задается двумя векторами. Запишем их в базисе +$(e_1,e_2,e_3,e_3)$: +$(a_1e_1+a_2e_2+a_3e_3+a_4e_4)\wedge(b_1e_1+b_2e_2+b_3e_3+b_4e_4)$. Обозначим +координату тензора $x$ при бивекторе $e_i\wedge e_j$ через +$x_{ij}$. Тогда разложимость $x$ равносильно обращению в $0$ выражения +$x_{12}x_{34}-x_{13}x_{24}+x_{14}x_{23}$. Это следует, например, из +соотношения на миноры матрицы +$\begin{pmatrix}a_1&a_2&a_3&a_4\\b_1&b_2&b_3&b_4\end{pmatrix}$. +\end{example} + +\subsection{Параболические подгруппы} +Оказывается, любое $X$, являющееся орбитой в $\mathbb P(V)$, задается +квадратичными уравнениями в проективных координатах. + +Пусть $v\in V$, $X=G\cdot\la v\ra$~--- орбита вектора +$v$. Тогда $\Stab_G(\la v\ra)=P$~--- параболическая подгруппа +в $G$. Проективное однородное многообразие задается подгруппой $P$ с +точностью до сопряженности. + +Посмотрим, как тор $T$ в $G$ действует на вектор $v$. Из равенства +$T\la v\ra=\la v\ra$ следует, что найдется $\lambda\colon T\to\mathbb +G_m$ (\term{вес} неприводимого представления $V$) такое, что +$tv=\lambda(t)v$ для всех $t\in T$. Представление задается своим старшим весом +(точнее, орбитой веса относительно $W$, но в этой орбите есть +единственный доминантный вес). %??? +% $v$~--- вектор старшего веса. +Пусть $\alpha_1,\dots,\alpha_l$~--- простые корни. +Рассмотрим базис $\alpha_1^\vee,\dots,\alpha_l^\vee$ +в двойственном пространстве, где $\alpha_i^\vee$ определяется +равенством +$\alpha_i^\vee(\beta) = \frac{2(\alpha_i,\beta)}{(\alpha_i,\alpha_i)}$. +Пусть $\varpi_1,\dots,\varpi_l$~--- двойственный к нему базис. +Таким образом, +$\frac{2(\alpha_i,\varpi)}{(\alpha_i,\alpha_i)} = \delta_{ij}$. +Эти элементы $\varpi_1,\dots,\varpi_l$ называются +\term{фундаментальными весами}. +Вес $\lambda$ раскладывается по этому базису следущим образом: +$\lambda = \sum m_i\omega_i$. +После этого $X$ (и $P$) зависит только от того, какие из $m_i$ не равны $0$. +То есть, проективные однородные многообразия задаются подмножеством +вершин на диаграмме Дынкина, состоящим из тех вершин, для которых +$m_i\neq 0$. +Мы будем их обводить на картинке. + +Например, картинка для проективного пространства такая: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (3p) at ($\sm*(3.3, 0)$); +\coordinate (d1) at ($\sm*(3.5, 0)$); +\coordinate (d2) at ($\sm*(3.7, 0)$); +\coordinate (d3) at ($\sm*(3.9, 0)$); +\coordinate (4m) at ($\sm*(4.1, 0)$); +\coordinate (4) at ($\sm*(4.6, 0)$); +\coordinate (5) at ($\sm*(6.0, 0)$); +\node at (1) [below=3pt,font=\scriptsize] {$1$}; +\node at (2) [below=3pt,font=\scriptsize] {$2$}; +\node at (3) [below=3pt,font=\scriptsize] {$3$}; +\node at (4) [below=3pt,font=\scriptsize] {$n-2$}; +\node at (5) [below=3pt,font=\scriptsize] {$n-1$}; +\draw (1)--(2); +\draw (2)--(3); +\draw (3)--(3p); +\draw (4m)--(4); +\draw (4)--(5); +\foreach \point in +{1,2,3,4,5} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\draw [black] (1) circle (5.0pt); +\foreach \point in +{d1,d2,d3} + { + \fill [black] (\point) circle (0.7pt); + } +\node (c) at ($\sm*(7.5, 0)$) {$\mathsf{A}_{n-1}$}; +\end{tikzpicture} +\] +Это соответствует векторному представлению $V = V(\varpi_1)$ группы +$\SL_n$. +Вообще, если на диаграмме $\mathsf{A}_{n-1}$ обвести вершину с номером $k$, +получится $\Gr(k,n)$. +Есть еще, например, присоединенное представление: $\SL_n$ действует +на своей алгебре Ли $\Lie(\SL_n)$. Картинка для этого представления +такая: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (2p) at ($\sm*(1.9, 0)$); +\coordinate (d1) at ($\sm*(2.1, 0)$); +\coordinate (d2) at ($\sm*(2.3, 0)$); +\coordinate (d3) at ($\sm*(2.5, 0)$); +\coordinate (3m) at ($\sm*(2.7, 0)$); +\coordinate (3) at ($\sm*(3.2, 0)$); +\coordinate (4) at ($\sm*(4.6, 0)$); +\node at (1) [below=3pt,font=\scriptsize] {$1$}; +\node at (2) [below=3pt,font=\scriptsize] {$2$}; +\node at (3) [below=3pt,font=\scriptsize] {$n-2$}; +\node at (4) [below=3pt,font=\scriptsize] {$n-1$}; +\draw (1)--(2); +\draw (2)--(2p); +\draw (3m)--(3); +\draw (3)--(4); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\draw [black] (1) circle (5.0pt); +\draw [black] (4) circle (5.0pt); +\foreach \point in +{d1,d2,d3} + { + \fill [black] (\point) circle (0.7pt); + } +\node (c) at ($\sm*(6.1, 0)$) {$\mathsf{A}_{n-1}$}; +\end{tikzpicture} +\] +Первая вершина соответствует $V$, последняя~--- $V^*$, +в итоге получаем $V^*\otimes V\isom \End(V)$. + +Если обведена одна вершина ($V = V(\varpi_i)$), то $P$ называется +\term{максимальной} параболической. +Если все вершины обведены, то $P$ называется \term{борелевской} +(это минимальная среди параболических). +Любая гладкая замкнутая подгруппа, содержащая $B$, называется +\term{параболической} и получается таким образом: $B\leq P \leq G$. + +Пусть теперь на диаграмме Дынкина системы $\mathsf{A}_{n-1}$ обведены +вершины с номерами $k_1,\dots,k_m$. +Полученное многообразие можно описать в терминах стандартного +представления $V = K^n$ группы $\SL_n$. +А именно, +\[ +X = \{U_1\leq\dots\leq U_m\mid \dim U_i = k_i\}. +\] +Такое $X$ называется \term{многообразием флагов}. +При этом $\SL_n$ действует на $X$ транзитивно. +Заметим, что тензорное произведение +$V(\varpi_{k_1})\otimes\dots\otimes V(\varpi_{k_m}0$ уже не обязано +быть неприводимым, но можно взять кусок, соответствующий +весу $\varpi_{k_1} + \dots + \varpi_{k_m}$. + +Так мы описали все однородные проективные многообразия для группы +$\SL_n$. +В общем случае (для произвольной $G$) иногда однородное проективное +многообразие называют \term{обобщенным флаговым многообразием}. +Его можно описать так: +\[ +X = \{P'\leq G\mid P'\mbox{ сопряжена с }P\}, +\] +где значок $P'\leq G$ означает, что $P'$~--- гладкая замкнутая подгруппа +в $G$. +Более точно, +\[ +X(R) = \{P'\leq G_R \mid\mbox{существуют }S/R, g\in G(S):\; gP'g^{-1} = P\}. +\] +После подкрутки на торсор $E$ получаем +\[ +{}_{E}X = \{P'\leq {}_{E}G\mid P'_{\ol{K}}\mbox{ сопряжена с }P_{\ol{K}} +\mbox{ внутри }({}_{E}G)_{\ol{K}} = G_{\ol{K}}\}. +\] +Обратите внимание, что в ${}_{E}G$ никакой $P$ может не оказаться. + +Проективное однородное многообразие $X$ \term{изотропно}, +если $X(K)\neq\emptyset$. +Сама группа ${}_{E}G$ называется \term{изотропной}, если для какого-то +проективного однородного многообразия ${}_{E}X$, отличного от точки, +${}_{E}X$ изотропно. + +Пусть $G = \PGL_n$. +Ее скрученная форма ${}_{E}G$ имеет вид $\Aut(A)$, а соответствующая +скрученная форма проективного пространства~--- $\SB(A)$. +Заметим, что у $\PGL_n$ (в отличие от $\SL_n$) нет векторного представления. +Почему? +Для начала поймем, откуда берется скрученная форма $\SL_n$. +Отображение определителя $\det\colon\GL_n\to\mathbb{G}_m$ +скручивается в \emph{приведенную норму} (\emph{reduced norm}) +\[ +\Nrd\colon A^* = \GL_1(A) \to \mathbb{G}_m. +\] +Ядро этого отображения обозначается через +$\SL_1(A) = \{g\in A\mid\Nrd(g)=1\}$. +Например, $(\Nrd(x))^n = \det(y\mapsto xy)$. + +Решетка корней содержится в решетке весов: +\[ +\mathbb{Z}\alpha_1\oplus\dots\oplus\mathbb{Z}\alpha_l +\leq +\mathbb{Z}\varpi\oplus\dots\oplus\mathbb{Z}\varpi_l +\] +Диаграммы Дынкина классифицируют расщепимые полупростые группы с точностью +до изогении, а класс изоморфности внутри класса изогении задается +промежуточной решеткой между этими двумя (с точностью до внешних +автоморфизмов). +Минимальная решетка $\mathbb{Z}\alpha_1\oplus\dots\oplus\mathbb{Z}\alpha_l$ +соответствует присоединенной группе (без центра); +максимальная решетка $\mathbb{Z}\varpi_1\oplus\dots\oplus\mathbb{Z}\varpi_l$ +соответствует односвязной группе (у нее самый большой центр). + + + +\end{document}