From b0946e45724648795a28dc7482c0eb90bcb7c302 Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Tue, 14 Jun 2016 23:49:00 +0300 Subject: [PATCH] Add Lecture 5 --- motives.pdf | Bin 502335 -> 551175 bytes motives.tex | 423 +++++++++++++++++++++++++++++++++++++++++++++++++++- 2 files changed, 422 insertions(+), 1 deletion(-) diff --git a/motives.pdf b/motives.pdf index 754e60756e2ef294956ce140829c1490bd219d12..90637c129f5bc08654fc7ac8f45d4240a4b3cbbb 100644 GIT binary patch delta 265637 zcmV)MK)AoZ)E<_ z$sC3h0X4HwF@hEWHM7%4+Y16WFq2U+6SKop+Z2-;iYN{;HVQ9HWo~D5Xdp5&Fq2S< zD}P&W8@UyJ*RLS@lstHbhj%WF1-#j28=z@}I4#;HMpv=bjhjRYWI6Hnp})P~8O=x{ zXGTF{)6ZX3`>@exj@g&e#f`c%{T5dHuQL<-+4N#Hb zNhgWpWjHo)E@V0b-lUr_(S-2EK`Oy$$L>}LqYE5dbRs(*vGvmP91DlQWu&9k$J`Sj3+Y7jLZtSNP6Gj%I+f+_*g#siH`-kRQ=<{1EeKBX>BsI)6+~Pepw$ z>Yo;`7NY)9Ji3`(uC5pJiXMyazn?sMqrJn&zmd|skxH#w*R{57O*-~nE4$~9`+{!Y zZe$Hh({1Z@%S9&KiFARh8scg)-SMvBCS~^A>`2AOwC{Cl>t>!;?N}$zi89}8eAn_+ zt#yX7`IK7kQ|o(r`E2a+d4E2$H;{-`UfKRR4@|6WEP0vbs#$W7Zd6I?BrKy-3v3c(^<^NBq zp-rKLx?uT&Q|w@&j&N_if%4C^@hzRnLPUdBZM7|h%GIx;qWJ7H2Q_FKhkN(tZHW=CAj_y9m}Lfy)ZG6eN2P?CeW?`CXxEj>D{UUYUQs zviDbLJF+)C?69JS_k{Oz)whAkp38pI^=;RM{^4Dy2KR*>n)R8g163W|?P@!Ao=9_x zGKcq0YbKqSZ+}Q0)T-7by?2M@_7-LD;B_S4=#d4wu&H-?WLq-qzWhMeKjh~5uL{u6 z$)kdG>LBZ~g>|#Q%Ymc!?(ZFkZG!YYBK-|v+|yqE>lh9lE&4ux%|Ad(@lw=JuU8jr z>*?vFe)a38nW$e(f0<3{CyV)NHecNcoO+pZQoo$tEPrm7m$RGZH|gfl^V!w)^oPZr zcrEe3Mq4TGL@+q%#)}C!Ex|&NYGh4XDZA4D_V<4X&p*6N%n;ZY4Hu98M7xdnuy) zF)31?&F2eneBIn~uu6Q%v2NG)W|}ngllq6n@_%Z!Y(&V5`VaN9`UwW`*aa(f32S1g zocF>STUqF4JkI-??^=9iJZo}(`*sDE^&hY2@9VSE(?&pj)*wQCUjOCgGyXh!w_1Jr zuCBj)`BHtHuHN0w888;hU+QO1p7V!da23_hr`PlP`tsv?SVbZ*S_G<@MFO>2f-+KhBo#Zk7%~5qN2ll(bG#Z)GKqsHzcYKnb+aAZ7j6#q9Iq>gN5g)$_AI z-5(bKXjtuqV~vdy8WyM+g>zJq`}*>_x~r}h*Nyt%ggu6@qYo$1oS=ImqX{SWpMO59 zHti3pwuEeaS#=e)GQ>gT*iT|3Nm}558bGW(Yr+n;YD+Y~kZr#IV#u3={9U`Lyz& zEKSJ9vos@fMpp)IFVR_~m6NJSXMeD9h;a9}O26^-CW-NQGVSK#=(aRjNRk}cS)-25 z5#P%ZYww#Q?ywwjY48EII+`M##a=47GC%8V6_hU1X;*oB06*eIqRh`2S!o-JbQ+sm ztz3~#1zkn2igXmP>>697i>9LRTW`BY&s|Adev$(#GA^|e9NZ~gb z0W!0I`UVvNGq;Bi0bUaUG?P#<6u0^!0Ra{PHMjgY0pJ`MH8mhGAa7!73OqatFHB`_ zXLM*FHZU|XlTnH&f89OHl3llPdw#{uZl=Tw;r$@Ps45BRR5p&vT$$p+WjUh4h$Sj6 z#s2$r#aga!ZJl&s32rz!S;><68NRa(aeLS;qyYfO)%hK20b^_Xgwy z*ZrJGICWR{+Jjqi#XOu!H}^SEK>E7`F5I>JTcl1!f1(aT!Z}rcBc#*@Zb2MTm&o75 z%Ul<>qCjbPz*G{*;s)~m^OnU58I`ikdIjSxGe)NI>VOs?T9`1yCaeI7N16B|BUh0! z_*+KX@QPF!FB5r?5M{14l;7G4&eb(xmX_&LBwFQCd7lW$BWqqb3vg8u{74^R84t3s zAJ2!Nf0V}ERfd44wd1ZP{fv^ej?@um_^HxvcwEjUNN}W;bp@(L zXVZ!2ZdJ{+4Hq)#uiKTXR2o;J7l{&3O4z&y@QW2j$^dgc1S2`e$eloFy3j5{1_V54 zn)4`Gm;gHNc9hg34r#j5E>1o$qXrP8v

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Из этого (а также из второго условия) следует, что $[D] = 0$ в $\Br(F)$. -\subsection{Пример: квадрика} +\subsection{Пример: квадрика}\label{ssect:quadric} Рассмотрим квадрику $Q = \{q=0\}$. В $Q\times Q = \{(\la u\ra, \la v\ra)$ есть подмножество $\{f(u,v)=0\}$, @@ -1664,4 +1669,420 @@ x_1 = x_2 = y_3 = 0. Почему-то это условие равносильно $x_3 = y_3 = 0$. \end{example} +% 19.03.2012 + +\subsection{Нерасщепимая квадрика} + +Что произойдет, если взять нерасщепимую квадрику? +Возьмем торсор $E\in H^1(F, O_{2n+2})$ и построим ${}_{E}Q$. +Как вычислить $\CH^*({}_{E}Q)$? +Более простой вопрос: +рассмотрим отображение +\[ +\CH^*({}_{E}Q) \xrightarrow{res} \CH^*(({}_{E}Q)_{\ol{F}}). +\] +Что можно сказать про образ этого отображения (то есть, +про рациональные циклы)? +Продолжим считать для простоты, что $n$ четно. +Мы знаем, что стоит в правой части: образующие +$1,h,h^2,\dots$ в коразмерностях $0,1,2,\dots,$ до середины, +образующие $[\pt],[\mathbb{P}^1],\mathbb{P}^2],\dots$ +в коразмерностях $n,n-1,n-2,\dots$ до середины, +и две образующие $[\Pi_1],[\Pi_2]$ в коразмерности $n/2$. +При этом $h^{n/2} = [\Pi_1] + [\Pi_2]$. +Умножение выглядит так: +$h\cdot[\mathbb{P}^i] = [\mathbb{P}]^{i-1}$, +$h\cdot[\Pi_1] = h\cdot [\Pi_2] = [\mathbb{P}^{n/2}]$. + +Во всяком случае, $h$ рационален: можно взять любую гладкую +подквадрику коразмерности $1$. +Пусть ${}_{E}Q$ задается уравнением $q=0$. +В случае расщепимой [четномерной] квадики это было уравнение +$x_1y_1 + \dots + x_{n/2+1}y_{n/2+1} = 0$, +и подквадрика выделялась дополнительным условием $x_{n/2+1} - y_{n/2+1}=0$. +В общем случае можно взять любой $v$ такой, что $q(v)\neq 0$, +и $q|_{\la v\ra^{\perp}}$ задает гладкую подквадрику коразмерности $1$. + +\begin{theorem}[Springer] +Предположим, что $q$ \term{анизотропна}, то есть, +$q(v)\neq 0$ при $v\neq 0$. +Тогда класс $[\pt]$ не рационален. +\end{theorem} + +Теорема доказывается так: класс $[\pt]$ рационален тогда и только тогда, +когда найдутся расширения $E_i/F$ такие, что +$\gcd([E_i:F]) = 1$, и над каждым $E_i$ квадрика +$q_{E_i}$ изотропна. +В частности, среди степеней расширений должна быть хотя бы одна нечетная, +и потому для некоторого $E/F$ с нечетным $[E:F]$ квадрика $q_E$ +изотропна. +Но из этого следует, что $q$ изотропна (это, собственно, и есть +классическая теорема Спрингера). + +Вот ответ на вопрос про образ: если $Q$ анизотропна, +то +\[ +\im(\CH^k({}_{E}Q) \to \CH^k(({}_{E}Q)_{\ol{F}})) += \begin{cases} +\CH^k(({}_{E}Q)_{\ol{F}}), & k < n/2, \\ +2\cdot\CH^k(({}_{E}Q)_{\ol{F}}), & k > n/2, \\ +2\cdot\mathbb{Z}[\Pi_1] + \mathbb{Z}([\Pi_1] - [\Pi_2]), & k = n/2 +\end{cases} % Check the coefficients here!! +\] +Если $q$ \term{изотропна}, то есть существует ненулевой вектор $v$ +такой, что $q(v) = 0$, +то можно выделить гиперболическую плоскость: +$q = \la 1,-1\ra \perp q'$. +Проитерируем этот процесс: получим +\[ +q = k\cdot \la 1, -1\ra \perp q_{\an}, +\] +где $q_{\an}$ и $k$ определены однозначно ($q_{\an}$~--- с точностью +до изометрии). +При этом $k$ называется \term{индексом Витта} формы $q$, +а $q_{\an}$~--- ее \term{анизотропной частью}. + +Так вот, если индекс Витта нашей формы $q$ равен $k$, +то циклы $[\pt], [\mathbb{P}^1], \dots, [\mathbb{P}^{k-1}]$ +рациональны. +Обратное тоже верно: если эти циклы рациональны, то индекс Витта +не меньше $k$. + +Резюме: рациональные циклы на самой квадрике контролируют только +ее индекс Витта. + +Посмотрим теперь на другое многообразие, связанное с торсором +$E\in H^1(F, O_{2n+2})$ (мы для удобства изменим нумерацию). +А именно, рассмотрим $\OGr(2,Q)$~--- многообразие вполне изотропных +плоскостей. +Мы реализовали $Q$ как $\{\la v\ra\mid q(v) = 0\}$. +Тогда $\OGr(2,Q) = \{\la u,v\ra\mid q(u) = q(v) = f(u,v) = 0\}$. + +Чтобы добраться до этого многообразия, +положим $X = \{f(u,v)=0\}$ +и рассмотрим фильтрацию из раздела~\ref{ssect:quadric}: +\[ +\begin{tikzcd} +Q\times Q & \arrow{d}{\mathbb{A}^{n}} & +\{f(u,v) = 0\} \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^1} & +\pt,\arrow[left hook->]{ll}\\ + & Q & & \OGr(1,2;Q). +\end{tikzcd} +\] +С одной стороны, $\OGr(1,2;Q)$~--- расслоение над $\OGr(2;Q)$ со слоем +$\mathbb{P}^1$. +С другой стороны, написанная фильтрация позволяет нам написать разложение +\[ +\CH^*(Q\times Q) = \CH^{*-n}(Q)\oplus\CH^{*-n+1}(\OGr(1,2;Q))\oplus\CH^*(Q). +\] +Более того, морфизмы в левую части из слагаемых в правой части задаются +явным образом (с помощью пулбэков и пушфорвардов), и они +$O(q)$-эквивариантны. + +\section{Мотивы Чжоу} + +\subsection{Категория соответствий} + +До сих пор мы смотрели на $\CH^*$ и на морфизмы вида +$\CH^*(X)\to\CH^*(\ol{X})$. +Посмотрим теперь на \term{мотив Чжоу} многообразия $X$. + +Начнем с категории гладких проективных многообразий над $F$. +Что в ней плохо? +Например, то, что морфизмы нельзя складывать: она не аддитивна +(и тем более не абелева). +Каждому морфизму $f\colon X\to Y$ можно сопоставить +его график $\Gamma_f\subseteq X\times Y$ +и получить $[\Gamma_f] \in \CH^*(X\times Y)$. +Элементы $\CH^*(X\times Y)$ уже можно складывать! +Поэтому в качестве промежуточного шага можно рассмотреть +\term{категорию соответствий} $\Cor_F$. +Ее объекты~--- гладкие проективные многообразия над $F$. +Морфизмы: $\Mor(X, Y) = \CH^{\dim Y}(X\times Y)$. +Роль тождественного морфизма играет класс диагонали. + +\begin{remark} +В этой конструкции можно заменить $\CH$ на что-то другое, +где есть пулбэки и пушфорварды (они понадобятся нам ниже), +например, на другую теорию когомологий. +Если взять $K$-теорию~--- получим \emph{$K$-мотивы}, +а не мотивы Чжоу). +\end{remark} + +Как определить композицию таких морфизмов? +Пусть $\alpha\in\CH^{\dim Y}(X\times Y)$, $\beta\in\CH^{\dim Z}(Y\times Z)$. +Рассмотрим диаграмму +\[ +\begin{tikzcd} +& X\times Y\times Z \arrow{dl}[swap]{\pr_{XY}} \arrow{dr}{\pr_{YZ}} +\arrow{dd}{\pr_{XZ}} \\ +X\times Y & & Y\times Z\\ +& X\times Z +\end{tikzcd} +\] +Из нее получается следующая диаграмма на уровне Чжоу: +\[ +\begin{tikzcd} +& \CH^*(X\times Y\times Z) +\arrow{dd}{(\pr_{XZ})_*} \\ +\CH^{\dim Y}(X\times Y)\arrow{ur}{\pr_{XY}^*} +& & \CH^{\dim Z}(Y\times Z)\arrow{ul}{\pr_{YZ}^*}\\ +& \CH^{*}(X\times Z) +\end{tikzcd} +\] +Поэтому +$\pr_{XY}^*(\alpha)\cdot\pr_{YZ}^*(\beta) +\in\CH^{\dim Y + \dim Z}(X\times Y\times Z))$, и +мы можем определить +\[ +\beta\circ\alpha = (\pr_{XZ})_*(\pr_{XY}^*(\alpha)\cdot\pr_{YZ}^*(\beta)) +\] +Это произведение имеет характер свертки. +Например, можно взять в качестве $X,Y,Z$ метрические пространства, +а в качестве морфизмов~--- ядерные операторы, +и получится свертка. +Или в качестве $X,Y,Z$~--- конечные множества, а в качестве морфизмов~--- +матрицы, и тогда получится произведение матриц. + +\subsection{Карубизация} + +Итак, в категории соответствий $\Cor_F$ морфизмы уже можно складывать: +это аддитивная категория. +Заметим, что в ней есть и прямые сумммы +($X\oplus Y = X\coprod Y$), и произведения ($X\otimes Y = X\times Y$). +Но эта категория не абелева (и даже не псевдоабелева). +Напомним, что категория называется \term{псевдоабелевой}, если +у любого проектора есть образ (в категорном смысле). +То есть, если $p\colon X\to X$~--- морфизм, для которого $p^2=p$, +то $X = X_1\oplus X_2$, причем $p$~--- проекция на $X_1$. + +Есть стандартная процедура, как из аддитивной категории получить +псевдоабелеву: \term{пополнение по Каруби} (\term{карубизация}). +Таким образом по $\Cor_F$ строится +\term{категория мотивов Гротендика--Чжоу} $\mathcal{M}$. +Ее объекты~--- пары $(X,p)$, где $p\colon X\to X$~--- идемпотент. +Неформально говоря, эта пара символизирует <<образ>> морфизма $p$ +(которого может не быть в исходной категории). +Морфизмы определяются так: +\[ +\Mor((X,p),(Y,q)) = q\circ\Mor(X,Y)\circ p. +\] +Есть функтор $\Cor\to\mathcal{M}$, $X\mapsto (X,\id_X)$. +Для многообразия $X$ объект $M(X) = (X,\id_X)$ называется +\term{мотивом $X$}. +На самом деле, нужно писать $\Cor_{\operatorname{rat},\operatorname{eff}}$ +вместо $\Cor$, и $\operatorname{Chow}^{\operatorname{eff}}$ вместо +$\mathcal{M}$. +Мы получили функторы +\[ +\begin{tikzcd} +\operatorname{SmProj}/F \arrow{r} \arrow[bend right=15, swap]{rr}{M} +& \Cor_{\operatorname{rat},\operatorname{eff}}(F) \arrow{r} +& \operatorname{Chow}^{\operatorname{eff}}(F), +\end{tikzcd} +\] +где $M$~--- функтор <<взятия мотива>>. +При этом $M(X\coprod Y) = M(X) \oplus M(Y)$, +$M(X\times Y) = M(X)\otimes M(Y)$. + +\subsection{Мотив проективной прямой} + +Попробуем <<посчитать>> мотив проективной прямой $M(\mathbb{P}^1)$. +Напомним, что у $\CH^(\mathbb{P}^1)$ стоит $\mathbb{Z}$ в коразмерностях +$0$ (с образующей $1$) и $1$ (с образующей $[\pt]$). +Рассмотрим вложение $i\colon \pt\to \mathbb{P}^1$ +и проекцию $\pi\colon\mathbb{P}^1\to\pt$. +Композиция $\pi\circ i \colon \pt \to \mathbb{P}^1 \to \pt$ +тождественна, поэтому $p = i\circ\pi$ является проектором на $\mathbb{P}^1$. +Это идемпотент, отправляющий все в точку. +Поэтому в категории мотивов +$M(\mathbb{P}^1) = M(\pt) \oplus (\mathbb{P}^1, 1 - [p])$. +Слагаемое $(\mathbb{P}^1, 1 - [p])$ обозначается через $\mathbb{L}$ +и называется \term{мотивом Лефшеца}. +Это аналог аффинной прямой в категории мотивов. +Оказывается, мотив Лефшеца неразложим. + +Мотив точки часто обозначается через $\mathbb Z = M(\pt)$; +он играет роль нейтрального объекта относительно $\otimes$. +При этом мотив Лефшеца $\mathbb{L}$ обозначается +через $\mathbb{Z}(1)[2] = \mathbb{Z}\{1\}$. +Тензорные степени мотива Лефшеца обозначаются так: +$L^{\otimes k} = \mathbb{Z}(k)[2k] = \mathbb{Z}\{k\}$. +Это в некотором смысле <<мотив>> $k$-мерного аффинного пространства. + +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. +Поэтому удобно обозначение +$M(Y)\otimes\mathbb{L}^{\otimes k} = M(Y)(k)[2k] = M(Y)\{k\}$. + +Что дает такого рода разложение? +\begin{fact} +Пусть +$M(X) = \bigoplus M(Y_i)\otimes\mathbb{L}^{\otimes k_i}$. +Тогда +$\CH^n(X) = \bigoplus\CH^{n-k_i}(Y_i)$. +\end{fact} +Например, из разложения $M(\mathbb{P}^1) = M(\pt)\otimes M(\pt)\{1\}$ +следует, что +\begin{align*} +\CH^0(\mathbb{P}^1) &= \CH^0(\pt) = \mathbb{Z},\\ +\CH^1(\mathbb{P}^1) &= \CH^1(\pt)\oplus\CH^0(\pt) = \mathbb{Z}. +\end{align*} +Вообще, $\CH^n(X) = \Mor(X,\mathbb{L}^{\otimes n})$ +и $\CH_n(X) = \Mor(\mathbb{L}^{\otimes n}, X)$, +то есть, $\CH$~--- представимый функтор в категории мотивов, +а $\mathbb{L}$ играет роль пространства Эйленберга--Маклейна. + +Умножение в $\CH^*$ тоже происходит из категории мотивов. +Пусть $\alpha\in\CH^k(X)$, $\beta\in\CH^n(X)$, то есть +$\alpha\colon M(X) \to \mathbb{L}^{\otimes k}$, +$\beta\colon M(X) \to \mathbb{L}^{\otimes n}$. +Перемножая эти отображения, получаем +\[ +\alpha\otimes\beta \colon M(X)\otimes M(X) \to +\mathbb{L}^{\otimes k} \otimes \mathbb{L}^{\otimes n}. +\] +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] +Пусть дана фильтрация многообразия $X$ замкнутыми (не обязательно гладкими) +подмножествами +\[ +\begin{tikzcd}[column sep=1.2em] +X = X_0 & \arrow{d}{\mathbb{A}^{k_0}} & +X_1 \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^{k_1}} & +X_2 \arrow[left hook->]{ll} & & \dots \arrow[left hook->]{ll} & & +X_n \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^{k_n}} & +X_{n+1} = \emptyset. \arrow[left hook->]{ll} \\ +& Y_0 & & Y_1 & & & \dots & & & Y_n +\end{tikzcd} +\] +Вертикальные стрелки означают, что для каждого $i=0,\dots,n$ +задан плоский морфизм $X_i\setminus X_{i+1} \to Y_i$, +слои которого~--- аффинные пространства $\mathbb{A}^{k_i}$. + +Тогда $M(X) = \bigoplus M(Y_i)\{k_i\}$ +и, кроме того, +$M(X) = \bigoplus M(Y_i)\{\dim X - \dim Y_i - k_i\}$. +В частности, имеется функториальный (по $Z$) +изоморфизм $\CH^*(X\times Z) \isom \bigoplus\CH^{*-k_i}(Y_i\times Z)$. +\end{theorem} + +\begin{example} +Фильтрация +\[ +\begin{tikzcd} +\mathbb{P}^1 & \arrow{d}{\mathbb{A}^1} & \pt\arrow[left hook->]{ll}\\ +& \pt +\end{tikzcd} +\] +приводит к разложению $M(\pt) \oplus M(\pt)\{1\}$. +\end{example} + +\begin{example} +Фильтрация +\[ +\begin{tikzcd} +Q\times Q & \arrow{d}{\mathbb{A}^{\dim Q}} & +\{f(u,v) = 0\} \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^1} & +Q\arrow[left hook->]{ll}\\ + & Q & & \OGr(1,2;f) +\end{tikzcd} +\] +из раздела~\ref{ssect:quadric} +приводит к разложению +\[ +M(Q\times Q) = M(Q) \oplus M(\OGr(1,2;Q))\{1\}\oplus M(Q)\{\dim Q\}. +\] +\end{example} + +\begin{example} +Пусть на квадрике $Q$ есть рациональная точка (то есть, форма $q$ +изотропна). +Тогда $q = \la 1,-1\ra \perp q'$, и есть фильтрация +\[ +\begin{tikzcd} +Q & \arrow{d}{\mathbb{A}^{\dim Q}} & +X' \arrow[left hook->]{ll} & \arrow{d}{\mathbb{A}^1} & +\pt,\arrow[left hook->]{ll}\\ + & \pt & & Q' +\end{tikzcd} +\] +где $Q' = \{q'=0\}$. +Получаем разложение +\[ +M(Q) = M(\pt) \oplus M(Q')\{1\} \oplus M(\pt)\{\dim Q\}. +\] +На картинке это выглядит так: +\[ +\begin{tikzpicture}[scale=1.0,thin] +\def\sm{0.7} +\coordinate (1) at ($\sm*(0, 0)$); +\coordinate (2) at ($\sm*(1.4, 0)$); +\coordinate (2p) at ($\sm*(1.9, 0)$); +\coordinate (d1) at ($\sm*(2.1, 0)$); +\coordinate (d2) at ($\sm*(2.3, 0)$); +\coordinate (d3) at ($\sm*(2.5, 0)$); +\coordinate (3m) at ($\sm*(2.7, 0)$); +\coordinate (3) at ($\sm*(3.2, 0)$); +\coordinate (4) at ($\sm*(4.2, 1)$); +\coordinate (5) at ($\sm*(4.2, -1)$); +\coordinate (6) at ($\sm*(5.2, 0)$); +\coordinate (6p) at ($\sm*(5.7, 0)$); +\coordinate (e1) at ($\sm*(5.9, 0)$); +\coordinate (e2) at ($\sm*(6.1, 0)$); +\coordinate (e3) at ($\sm*(6.3, 0)$); +\coordinate (7m) at ($\sm*(6.5, 0)$); +\coordinate (7) at ($\sm*(7.0, 0)$); +\coordinate (8) at ($\sm*(8.4, 0)$); +\draw (1)--(2); +\draw (2)--(2p); +\draw (3m)--(3); +\draw (3)--(4); +\draw (3)--(5); +\draw (4)--(6); +\draw (5)--(6); +\draw (6)--(6p); +\draw (7m)--(7); +\draw (7)--(8); +\draw[dotted] ($\sm*(1, 1.3)$)--($\sm*(7.4, 1.3)$)--($\sm*(7.4,-1.3)$) +--($\sm*(1, -1.3)$)--($\sm*(1, 1.3)$); +\foreach \point in +{1,2,3,4,5,6,7,8} + { + \fill [white] (\point) circle (2.0pt); + \draw [black] (\point) circle (2.0pt); + } +\draw [black] (1) circle (5.0pt); +\draw [black] (8) circle (5.0pt); +\foreach \point in +{d1,d2,d3,e1,e2,e3} + { + \fill [black] (\point) circle (0.7pt); + } +\node at (1) [below=3pt,font=\scriptsize] {$1$}; +\node at (8) [below=3pt,font=\scriptsize] {$[\pt]$}; +\draw [|->] ($\sm*(0, -1)$) --node[below=3pt, font=\scriptsize] {$1$} +($\sm*(1.4, -1)$); +\draw [|->] ($\sm*(0, -2)$) --node[below=3pt, font=\scriptsize] {$\dim Q$} +($\sm*(8.4, -2)$); +\end{tikzpicture} +\] +Обратите внимание, что на картинке выделен мотив подквадрики $Q'$, +который сдвигается на $1$. +Кроме того, мотив точки (справа) сдвигается на $\dim Q$. +Иными словами, у нас появились проекторы +$1\times [\pt]$, $[\pt]\times 1$, $\Delta_Q - 1\times[\pt] - [\pt]\times 1$. +\end{example} + +\begin{remark} +Обозначение $\mathbb{Z}(1)[2]$ для мотива Лефшеца может показаться странным. +Здесь второй сдвиг соответствует сдвигу в триангулированной категории +Воеводского. +При желании можно представлять это как композицию двух сдвигов: +$(1)[1]$~--- сдвиг на $\mathbb{G}_m$, $(0)[1]$~--- сдвиг на $S^1$. +\end{remark} + \end{document}