From ca271db1b414bda9d6129419656d851908de3d52 Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Wed, 15 Jun 2016 23:33:43 +0300 Subject: [PATCH] Add Lecture 6 --- motives.pdf | Bin 551175 -> 579995 bytes motives.tex | 477 +++++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 474 insertions(+), 3 deletions(-) diff --git a/motives.pdf b/motives.pdf index 90637c129f5bc08654fc7ac8f45d4240a4b3cbbb..2f3d518e9488bb6db74d0f6c9d062533821dd398 100644 GIT binary patch delta 209834 zcmV)IK)k<)(jl9(BN8Q0L`E$!E;R}z(A34)3NbP;u@boq0yH(V%?#=f0X2~k6SK=4 zg%p=?g8>w?WHE0R0yi_W)koP2lb>iQ4L1reOl59obZ8(lGBcAwlPQ12T3d75#ua|& zuh{96{UBiPXE&qCjN(+OXX?~rIhnS$rlBP^Qf-qSin5hH^tbmrAV6{#00C1{G9wCD z;Ox1dZ_jdRg%m*;DWXu)38jUyQYa%-R6=>7wHJzn*Ggz9yu%YEg2Tv)5d1`Aj?zKk zPbqJN1|J=~FkoarYQTTV1V{iItBo)ul=E7!5En^UNQvZywcxMR#5xG@7=xvjpo@?M zrcUGPaD@nRQm`TvoNxq@G6*kmwMKY|RMrb`aSdwX9aO+14;5GsABAzA1Zk)Q{sff* zM+WAudm?pP-UI~*&eBn%~X4dUSlx+z4&MfeX+#*JfdBPz=A z6Ii9-as_vL4aJn!!W&H!xWT|Sg}_%iVQn+049M}*o3{?nHbXrX~ 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z?wv3;KCE@@IE`UI}%<+9cj yzTD+;HjJ^2k%J?#y|KRae?waX3m6ASeS1e&dt(zARu)c926h-yQV}^(nEwUsC=$y6 diff --git a/motives.tex b/motives.tex index d86f447..ace0aca 100644 --- a/motives.tex +++ b/motives.tex @@ -61,6 +61,7 @@ \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\pr}{pr} \DeclareMathOperator{\id}{id} +\DeclareMathOperator{\res}{res} %\DeclareFontFamily{OT1}{pzc}{} %\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [1.2] pzcmi7t}{} @@ -85,6 +86,8 @@ \newtheorem{example}[theorem]{Пример} \newtheorem{fact}[theorem]{Факт} \newtheorem{remark}[theorem]{Замечание} +\newtheorem{exercise}[theorem]{Упражнение} +\newtheorem{definition}[theorem]{Определение} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} @@ -92,6 +95,7 @@ \newcommand{\rAngle}{\rangle\!\rangle} \newcommand{\trleq}{\trianglelefteq} \newcommand{\ol}{\overline} +\newcommand{\wt}{\widetilde} \newcommand{\TBW}{\textbf{TBW}} @@ -1679,7 +1683,7 @@ x_1 = x_2 = y_3 = 0. Более простой вопрос: рассмотрим отображение \[ -\CH^*({}_{E}Q) \xrightarrow{res} \CH^*(({}_{E}Q)_{\ol{F}}). +\CH^*({}_{E}Q) \xrightarrow{\res} \CH^*(({}_{E}Q)_{\ol{F}}). \] Что можно сказать про образ этого отображения (то есть, про рациональные циклы)? @@ -1781,7 +1785,7 @@ $O(q)$-эквивариантны. \section{Мотивы Чжоу} -\subsection{Категория соответствий} +\subsection{Категория соответствий}\label{ssect:corr-category} До сих пор мы смотрели на $\CH^*$ и на морфизмы вида $\CH^*(X)\to\CH^*(\ol{X})$. @@ -1910,6 +1914,8 @@ $M(\mathbb{P}^1) = M(\pt) \oplus (\mathbb{P}^1, 1 - [p])$. $L^{\otimes k} = \mathbb{Z}(k)[2k] = \mathbb{Z}\{k\}$. Это в некотором смысле <<мотив>> $k$-мерного аффинного пространства. +\subsection{Представимость функтора Чжоу} + Часто удается разложить мотив многообразия $X$ в прямую сумму вида $M(X) = \bigoplus M(Y_i)\otimes\mathbb{L}^{\otimes k_i}$, где $Y_i$~--- какие-то другие многообразия. @@ -1946,7 +1952,9 @@ $\beta\colon M(X) \to \mathbb{L}^{\otimes n}$. Правая часть изоморфна $\mathbb{L}^{\otimes(k+n)}$. Взяв композицию с морфизмом $M(\Delta)\colon M(X) \to M(X\times X)$, получаем $\alpha\cup\beta\colon M(X) \to \mathbb{L}^{\otimes(k+n)}$. -\begin{theorem}[Карпенко, 2000] + +\subsection{Теорема Карпенко} +\begin{theorem}[Карпенко, 2000]\label{thm:karpenko} Пусть дана фильтрация многообразия $X$ замкнутыми (не обязательно гладкими) подмножествами \[ @@ -1970,6 +1978,9 @@ $M(X) = \bigoplus M(Y_i)\{\dim X - \dim Y_i - k_i\}$. изоморфизм $\CH^*(X\times Z) \isom \bigoplus\CH^{*-k_i}(Y_i\times Z)$. \end{theorem} +Фильтрация из теоремы~\ref{thm:karpenko} называется +\term{относительным клеточным разложением}. + \begin{example} Фильтрация \[ @@ -2085,4 +2096,464 @@ $1\times [\pt]$, $[\pt]\times 1$, $\Delta_Q - 1\times[\pt] - [\pt]\times 1$. $(1)[1]$~--- сдвиг на $\mathbb{G}_m$, $(0)[1]$~--- сдвиг на $S^1$. \end{remark} +\begin{remark} +В общем случае, разложение Брюа показывает, что если $G$~--- расщепимая +группа, $P$~--- ее параболическая подгруппа, +то мотив однородного многообразия $G/P$ равен прямой сумме +сдвигов $\mathbb{Z}$: +$M(G/P) = \bigoplus\mathbb{Z}\{\dots\}$. +При этом $\mathbb{Z}\{i\}$ встречается столько раз, каково количество +минимальных представителей классов смежности из $W/W_P$ длины $i$. +Поэтому сдвиги считываются из диаграммы Хассе. +\end{remark} +\begin{example} +Например, для $\mathbb{P}^n$ диаграмма Хассе выглядит так: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (2p) at ($\sm*(1.9, 0)$); +\coordinate (d1) at ($\sm*(2.1, 0)$); +\coordinate (d2) at ($\sm*(2.3, 0)$); +\coordinate (d3) at ($\sm*(2.5, 0)$); +\coordinate (3m) at ($\sm*(2.7, 0)$); +\coordinate (3) at ($\sm*(3.2, 0)$); +\coordinate (4) at ($\sm*(4.6, 0)$); +\node at (1) [below=3pt,font=\scriptsize] {$0$}; +\node at (2) [below=3pt,font=\scriptsize] {$1$}; +\node at (3) [below=3pt,font=\scriptsize] {$n-1$}; +\node at (4) [below=3pt,font=\scriptsize] {$n$}; +\draw (1)--(2); +\draw (2)--(2p); +\draw (3m)--(3); +\draw (3)--(4); +\foreach \point in +{1,2,3,4} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{d1,d2,d3} + { + \fill [black] (\point) circle (0.7pt); + } +\end{tikzpicture} +\] +Поэтому мотив $\mathbb{P}^n$ равен +$\mathbb{Z}\oplus\mathbb{Z}\{1\}\oplus\dots\oplus\mathbb{Z}\{n\}$. +Для [расщепимой] квадрики $Q$ четной размерности диаграмма такая: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (2p) at ($\sm*(1.9, 0)$); +\coordinate (d1) at ($\sm*(2.1, 0)$); +\coordinate (d2) at ($\sm*(2.3, 0)$); +\coordinate (d3) at ($\sm*(2.5, 0)$); +\coordinate (3m) at ($\sm*(2.7, 0)$); +\coordinate (3) at ($\sm*(3.2, 0)$); +\coordinate (4) at ($\sm*(4.2, 1)$); +\coordinate (5) at ($\sm*(4.2, -1)$); +\coordinate (6) at ($\sm*(5.2, 0)$); +\coordinate (6p) at ($\sm*(5.7, 0)$); +\coordinate (e1) at ($\sm*(5.9, 0)$); +\coordinate (e2) at ($\sm*(6.1, 0)$); +\coordinate (e3) at ($\sm*(6.3, 0)$); +\coordinate (7m) at ($\sm*(6.5, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\coordinate (8) at ($\sm*(8.4, 0)$); +\node at (1) [below=3pt,font=\scriptsize] {$0$}; +\node at (2) [below=3pt,font=\scriptsize] {$1$}; +\node at (5) [below=3pt,font=\scriptsize] {$n/2$}; +\node at (7) [below=3pt,font=\scriptsize] {$n-1$}; +\node at (8) [below=3pt,font=\scriptsize] {$n$}; +\draw (1)--(2); +\draw (2)--(2p); +\draw (3m)--(3); +\draw (3)--(4); +\draw (3)--(5); +\draw (4)--(6); +\draw (5)--(6); +\draw (6)--(6p); +\draw (7m)--(7); +\draw (7)--(8); +\foreach \point in +{1,2,3,4,5,6,7,8} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{d1,d2,d3,e1,e2,e3} + { + \fill [black] (\point) circle (0.7pt); + } +\end{tikzpicture} +\] +Поэтому $M(Q) = \mathbb{Z}\oplus\mathbb{Z}\{1\}\oplus\dots +\oplus\mathbb{Z}\{n/2\}^{\otimes 2} \oplus\mathbb{Z}\{n/2+1\}\oplus\dots +\oplus\mathbb{Z}\{n\}$. +\end{example} + +\subsection{Метод общей точки} + +Пока что мы получали мотивные разложения только с помощью фильтраций +и теоремы Карпенко. +Сейчас мы узнаем еще один прием~--- \emph{метод общей точки}. +Пусть $X,Y$~--- многообразия, причем $Y$ неприводимо. +Рассмотрим $X_{F(Y)}$ и его кольцо Чжоу $\CH^*(X_{F(Y)}$ +(напомним, что $F(Y)$~--- поле рациональных функций на $Y$). +\begin{lemma} +Отображение $\CH^*(X\times Y)\to\CH^*(X_{F(Y)})$, +полученное из декартова квадрата +\[ +\begin{tikzcd} +X_{F(Y)} \arrow{r}\arrow{d} & X\times Y \arrow{d} \\ +\Spec F(Y) \arrow[right hook->]{r} & Y, +\end{tikzcd} +\] +сюръективно. +\end{lemma} +\begin{proof} +Указанную диаграмму можно представлять себе как индуктивный предел +диаграмм вида +\[ +\begin{tikzcd} +X\times U \arrow{r}\arrow{d} & X\times Y \arrow{d} \\ +U \arrow[right hook->]{r} & Y, +\end{tikzcd} +\] +где $U$~--- открытое непустое в $Y$ (поскольку $\Spec F(Y) = \varinjlim U$). +Но каждое полученное таким образом отображение +$\CH^*(X\times Y) \to \CH^*(X\times U)$ сюръективно в силу точной +последовательности локализации. +\end{proof} + +Как этим пользоваться? +Чтобы выделить прямое слагаемое в мотиве $X$, нам нужно +найти проектор $p\in\CH^{\dim X}(X\times X)$. +Для этого есть два варианта: +\begin{enumerate} +\item взять $Y:= X$, выбрать какой-то элемент из $\CH^i(X_{F(X)})$ +и поднять его в $\CH^i(X\times X)$; +\item взять какой-нибудь $Y$, построить элементы из $\CH^i(X\times Y)$, +$\CH^i(Y\times X)$, взять их композицию, и дальше как в первом пункте. +\end{enumerate} +Пусть $X$~--- гладкое проективное над $F$. +Напомним, что цикл $\alpha\in\CH^*(X_{\ol{F}})$ называется +рациональным, если он лежит в образе отображения +\[ +\res\colon\CH^*(X) \to \CH^*(X_{\ol{F}}). +\] +Аналогично, можно рассмотреть отображение +\[ +\res\colon\CH^*(X\times X) \to \CH^*(X_{\ol{F}}\times X_{\ol{F}}). +\] +Правая часть гораздо лучше левой. + +Предположим, что мы нашли цикл $p\in\CH^*(X_{\ol{F}}\times X_{\ol{F}})$ +такой, что +\begin{enumerate} +\item $p$~--- проектор (в смысле композиции, определенной в +разделе~\ref{ssect:corr-category}); +\item $p$ рационален, то есть, $p$ поднимается до какого-то +$\wt{p}\in\CH^*(X\times X)$. +\end{enumerate} +Следует ли из этого, что $\wt{p}$ является проектором? +Вообще говоря~--- нет, но для однородных многообразий есть такая теорема. + +\begin{theorem}[Теорема нильпотентности Роста] +Если $X$~--- проективное однородное многообразие, +$p$~--- рациональный проектор на $X_{\ol{F}}$, то он +поднимается до проектора $\wt{p}$ на $X$. +Более сильное утверждение: +\[ +\ker(\CH^*(X\times X) \to \CH^*(X_{\ol{F}}\times X_{\ol{F}})) +\] +состоит из нильпотентных (в смысле композиции) элементов. +\end{theorem} + +Как выглядят очевидные элементы $\CH^*(X\times X)$? +Можно взять $\alpha\in\CH^*(X)$, $b\in\CH^*(X)$, и получить +$a\times b\in\CH^*(X\times X)$. + +\begin{exercise} +В этом случае $(a\times b)\circ (c\times d) = \deg(ad)(c\times b)$, +где $\deg\colon\CH^*(Y) \to \CH^*(\pt)$ происходит из +морфизма $Y\to\pt$. +\end{exercise} + +\begin{exercise} +Пусть $R$~--- коммутативное кольцо, $I\trleq R$~--- идеал, состоящий +из нильпотентных элементов. +Тогда любой идемпотентв $R/I$ поднимается до идемпотента в $R$. +\end{exercise} + +\begin{definition} +Многообразие $X$ называется \term{клеточным}, если существует +фильтрация вида +\[ +\begin{tikzcd}[column sep=1.2em] +X = X_0 & \arrow{d}{\mathbb{A}^{k_0}} & +X_1 \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^{k_1}} & +X_2 \arrow[left hook->]{ll} & & \dots \arrow[left hook->]{ll} & & +X_n \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^{k_n}} & +X_{n+1} = \emptyset, \arrow[left hook->]{ll} \\ +& \pt & & \pt & & & \dots & & & pt +\end{tikzcd} +\] +в которой все базы~--- точки. +\end{definition} +\begin{example} +Разложение Брюа говорит, что если группа $G$ расщепима, +$P$~--- параболическая подгруппа в $G$, +то многообразия $G/P$ клеточное. +\end{example} +Из теоремы Карпенко~\ref{thm:karpenko} следует, что для клеточного +многообразия $M(X) = \bigoplus \mathbb{Z}\{r_i\}$. + +\begin{definition} +Многообразие $X$ называется \term{клеточным над общей точкой} +(\term{generically cellular}), если $X_{F(X)}$ клеточное. +\end{definition} +\begin{example} +Пусть $Q$~--- \term{Квадрика Пфистера}, то есть, $Q = \{q=0\}$, +где $q = \lAngle a_1,\dots,a_k\rAngle = \la 1,-a_1\ra \otimes\dots +\otimes\la 1,-a_k\ra$~--- $k$-кратная форма Пфистера +(размерность $Q$ равна $2^k-2$). +Тогда $Q$ клеточная над общей точкой. +\end{example} +Верно и обратное: все анизотропные четномерные квадрики, +клеточные над общей точкой, так выглядят. + +\subsection{Мотив квадрики Пфистера} + +Пусть $Q$~--- квадрика Пфистера размерности $2^k - 2$. +Мы знаем, что $Q_{F(Q)}$~--- клеточное многообразие. +Обозначим образующие компонент $\CH^*(Q_{F(Q)})$: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (2p) at ($\sm*(1.9, 0)$); +\coordinate (d1) at ($\sm*(2.1, 0)$); +\coordinate (d2) at ($\sm*(2.3, 0)$); +\coordinate (d3) at ($\sm*(2.5, 0)$); +\coordinate (3m) at ($\sm*(2.7, 0)$); +\coordinate (3) at ($\sm*(3.2, 0)$); +\coordinate (4) at ($\sm*(4.2, 1)$); +\coordinate (5) at ($\sm*(4.2, -1)$); +\coordinate (6) at ($\sm*(5.2, 0)$); +\coordinate (6p) at ($\sm*(5.7, 0)$); +\coordinate (e1) at ($\sm*(5.9, 0)$); +\coordinate (e2) at ($\sm*(6.1, 0)$); +\coordinate (e3) at ($\sm*(6.3, 0)$); +\coordinate (7m) at ($\sm*(6.5, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\coordinate (8) at ($\sm*(8.4, 0)$); +\node at (1) [below=3pt,font=\scriptsize] {$0$}; +\node at (2) [below=3pt,font=\scriptsize] {$1$}; +\node at (5) [below=3pt,font=\scriptsize] {$2^{k-1}-1$}; +\node at (7) [below=3pt,font=\scriptsize] {$2^k-3$}; +\node at (8) [below=3pt,font=\scriptsize] {$2^k-2$}; +\node at (1) [above=3pt,font=\scriptsize] {$1$}; +\node at (2) [above=3pt,font=\scriptsize] {$h$}; +\node at (4) [above=3pt,font=\scriptsize] {$\rho$}; +\node at (6) [above=3pt,font=\scriptsize] {$h\rho$}; +\node at (8) [above=3pt,font=\scriptsize] {$h^{2^{k-1}}\rho$}; +\draw (1)--(2); +\draw (2)--(2p); +\draw (3m)--(3); +\draw (3)--(4); +\draw (3)--(5); +\draw (4)--(6); +\draw (5)--(6); +\draw (6)--(6p); +\draw (7m)--(7); +\draw (7)--(8); +\foreach \point in +{1,2,3,4,5,6,7,8} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{d1,d2,d3,e1,e2,e3} + { + \fill [black] (\point) circle (0.7pt); + } +\end{tikzpicture} +\] +Возьмем $\rho\in\CH^*(Q_{F(Q)})$ и поднимем его до какого-то элемента +$\alpha\in\CH^*(Q\times Q)$. +Рассмотрим коммутативную диаграмму +\[ +\begin{tikzcd} +\alpha\in\CH^*(Q\times Q) \arrow[->>]{r} \arrow{d}{\res} +& \CH^*(Q_{F(Q)})\ni\rho \arrow{d}{\isom} \\ +\ol{\alpha}\in\CH^*(Q_{\ol{F}}\times Q_{\ol{F}}) \arrow[->>]{r} +& \CH^*(Q_{\ol{F}(Q)})\ni\ol{\rho} +\end{tikzcd} +\] +Мы не умеем следить за $\alpha$, но знаем, что $\ol{\alpha} = \res(\alpha)$ +переходит в $\ol{\rho}$ (который отождествляется с $\rho$ +при помощи изоморфизма), и знаем, как выглядит нижняя +горизонтальная стрелка. + +Итак, $\ol{\alpha}$ является прообразом $\rho$, поэтому обязан иметь вид +\[ +\ol{\alpha} = \ol{\rho}\times 1 ++ c_1\cdot h^{2^{k-1}-2} \times h ++ c_2\cdot h^{2^{k-1}-3} \times h^2 ++ \dots ++ c_{2^{k-1}-1}\cdot 1\times h^{2^{k-1}-1} ++ c\cdot 1\times\ol{\rho}. +\] +Заметим, что все слагаемые в правой части, кроме первого и последнего, +содержатся в образе $\res$, посольку $h$ рационален. +Поэтому (подправив $\alpha$) можно считать, +что $\ol\alpha = \ol\rho\times 1 + c\cdot 1 \times \ol\rho$. + +Кроме того, цикл $2\ol\rho$ рационален, поскольку квадрика $Q$ +ращепляется квадратичным расширением. +Действительно, если $q = \lAngle a_1,\dots,a_k\rAngle$, достаточно взять +поле $F(\sqrt{a_1})$. +Над этим полем $q$ изотропна, а потому гиперболична. +Рассуждение заканчивается рассмотрением диаграммы +\[ +\begin{tikzcd} +\CH^*(Q) \arrow[yshift=-2pt]{r} \arrow{d} & \CH^*(Q_{F(\sqrt{a_1})}) \arrow{d} +\arrow[yshift=2pt]{l} \\ +\CH^*(Q_{\ol{F}}) \arrow[yshift=-2pt]{r} & \CH^*(Q_{\ol{F}(\sqrt{a_1})}). +\arrow[yshift=2pt]{l} +\end{tikzcd} +\] + +Стало быть, либо $\ol\rho\times 1$ рационален, либо +$\ol\rho\times 1 + 1\times\ol\rho$ рационален (в зависимости от четности $c$). + +Предположим для начала, что $\ol\rho\times 1$ рационален. +Докажем, что в этом случае $Q$ изотропна. +Действительно, циклы $\ol\rho\times 1$ и $1\times\ol\rho$ рациональны, +а потому и $(\ol\rho\times 1)\cdot(1\times\ol\rho) = \ol\rho\times\ol\rho$ +рационален. +Кроме того, $\ol{h}\times 1$ и $1\times\ol{h}$ рациональны, +а потому и $\ol{\pt}\times\ol{\pt}$ рационален. +Рассмотрим пушфорвард относительно проекции $Q\times Q$ на первый сомножитель: +\begin{align*} +\CH^*(\ol{Q}\times\ol{Q}) &\to \CH^*(\ol{Q}), +\ol{\pt}\times\ol{\pt} &\mapsto \ol{\pt}. +\end{align*} +Поэтому и цикл $\ol{\pt}$ рационален. +Значит, на $Q$ есть $0$-цикл степени $1$. +По теореме Спрингера из этого следует, что на $Q$ есть рациональная точка, +то есть, $Q$ изотропна~--- и это неинтересный случай. + +Значит, на самом деле цикл $\ol\rho\times 1 + 1\times\ol\rho$ +рационален. +Из него можно постараться изготовить проектор. +Заметим, что для любых $i,j$ цикл +$(\ol{h}^i\ol\rho)\times \ol{j}^j + \ol{h}^i\times (\ol{h}^j\ol\rho)$ +тоже рационален. +Как подобрать $i,j$, чтобы это был проектор? +Заметим, что $\ol\rho$ лежит в коразмерности $2^{k-1} - 1$, +поэтому нужно, чтобы $j = 2^{k-1} - 1 - i$. +Оказывается, этого достаточно: нужно вспомнить формулу +$(a\times b)(c\times d) = \deg(ad)c\times b$ +и равенство $\ol{h}^{2^{k-1}-1}\ol\rho = \ol{\pt}$. +После этого прямое вычисление показывает, что +мы получили проектор. +Варьируя $i$, получаем $2^{k-1} - 1$ проекторов. + +Соответствующее разложение мотива $Q$ можно нарисовать так: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (3) at ($\sm*(2.8, 0)$); +\coordinate (3p) at ($\sm*(3.3, 0)$); +\coordinate (d1) at ($\sm*(3.5, 0)$); +\coordinate (d2) at ($\sm*(3.7, 0)$); +\coordinate (d3) at ($\sm*(3.9, 0)$); +\coordinate (4m) at ($\sm*(4.1, 0)$); +\coordinate (4) at ($\sm*(4.6, 0)$); +\coordinate (5) at ($\sm*(6.0, 0)$); +\coordinate (6) at ($\sm*(7.0, 1)$); +\coordinate (7) at ($\sm*(7.0, -1)$); +\coordinate (8) at ($\sm*(8.0, 0)$); +\coordinate (9) at ($\sm*(9.4, 0)$); +\coordinate (9p) at ($\sm*(9.9, 0)$); +\coordinate (e1) at ($\sm*(10.1, 0)$); +\coordinate (e2) at ($\sm*(10.3, 0)$); +\coordinate (e3) at ($\sm*(10.5, 0)$); +\coordinate (10m) at ($\sm*(10.7, 0)$); +\coordinate (10) at ($\sm*(11.2, 0)$); +\coordinate (11) at ($\sm*(12.6, 0)$); +\coordinate (12) at ($\sm*(14.0, 0)$); +\draw (1)--(2)--(3)--(3p); +\draw (4m)--(4)--(5)--(6); +\draw (5)--(7); +\draw (6)--(8); +\draw (7)--(8)--(9)--(9p); +\draw (10m)--(10)--(11)--(12); +\draw[dotted] (1) edge[bend left=45](6); +\draw[dotted] (2) edge[bend left=35](8); +\draw[dotted] (3) edge[bend left=45](9); +\draw[dotted] (4) edge[bend right=45](10); +\draw[dotted] (5) edge[bend right=35](11); +\draw[dotted] (7) edge[bend right=45](12); +\foreach \point in +{1,2,3,4,5,6,7,8,9,10,11,12} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\foreach \point in +{d1,d2,d3,e1,e2,e3} + { + \fill [black] (\point) circle (0.7pt); + } +\end{tikzpicture} +\] +Иными словами, над базой слагаемые в разложении мотива квадрики +объединяются в пары. + +\begin{exercise} +Пусть $R$~--- коммутативное кольцо, $I\trleq R$~--- идеал, состоящий +из нильпотентных элементов. +Тогда любой обратимый элемент $R/I$ поднимается до обратимого элемента в $R$. +\end{exercise} + +Слагаемое, которое дает первый проектор из этих, +называется \term{мотивом Роста} и обозначается через $R$. +Таким образом, над замыканием +$\ol{R} = \mathbb{Z} \oplus \mathbb{Z}\{2^{k-1}-1\}$. +Мотив квадрики Пфистера, таким образом, составлен +из сдвигов мотива Роста: +\[ +M(Q) = R \oplus R\{1\} \oplus \dots \oplus R\{2^{k-1}-1\}. +\] +\begin{fact} +Пусть $\lAngle a_1,\dots,a_k\rAngle$, +$\lAngle a'_1,\dots,a'_k\rAngle$~--- две $k$-формы Пфистера. +Соответствующие этим квадрикам мотивы Роста изоморфны (в категории мотивов) +тогда и только тогда, когда сами формы изоморфны, +что в свою очередь равносильно равенству чашечных произведений +$(a_1)\cup\dots\cup(a_k) = (a'_1)\cup\dots\cup(a'_k)$ +в $H^k(F, \mathbb{Z}/2\mathbb{Z})$. +\end{fact} + +\begin{remark} +Мотив Роста над замыканием превращается в +$\mathbb{Z} \oplus \mathbb{Z}\{2^{k-1}-1\}$. +Верно и обратное: +если мотив над замыканием выглядит так, то это мотив Роста +(теорема Никиты Семенова). +\end{remark} + \end{document}