\documentclass[10pt]{amsart} \usepackage[leqno]{amsmath} \usepackage{amssymb,mystyle,amscd,diagrams} \topmargin=-0.2in \oddsidemargin=-0.2in \evensidemargin=-0.2in \textwidth=6.9in \textheight=8.7in \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\Rad}{Rad} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\supp}{Supp} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\height}{ht} \DeclareMathOperator{\Char}{char} \DeclareMathOperator{\codim}{codim} \renewcommand{\Im}{{\rm Im}} \newcommand{\B}{\mathscr{B}} \newcommand{\I}{\mathscr{I}} \newcommand{\calC}{\mathscr{C}} \newcommand{\calV}{\mathscr{V}} \renewcommand{\a}{\mathfrak{a}} \newcommand{\G}{\mathbf G} \begin{document} \smallskip \centerline{\sc Math 632, Lecture 18\\ February 18, 2004} \section{Scheme theoretic intersection} If $i:Z\rightarrow X$ and $i':Z'\rightarrow X$ are closed immersions of ringed spaces and there exists a compatible map $f:Z\rightarrow Z'$ then $f$ is also a closed immersion. If in addition there exists a compatible $f':Z'\rightarrow Z$ then necessarily $f\circ f'=\id_{Z'}$ and $f'\circ f=\id_{Z}$. We say the two closed immersions $Z,Z'\rightarrow X$ are {\em equivalent} if there exists a compatible isomorphism $Z\simeq Z'$. Such a map is unique if it exists. \begin{definition} Let $Z,Z'\rightarrow X$ be closed immersions of schemes. The {\em scheme theoretic intersection} of $Z,Z'$ is \begin{diagram} Z\cap Z'=Z\times_X Z' & \rTo & Z'\\ \dTo & & \dTo \\ Z & \rTo & X\\ \end{diagram} and $Z\cap Z'$ is realized, via either projection, as a closed subscheme of $X$ whose image is contained in $Z$ and $Z'$. \end{definition} Observe that if $Y\rightarrow X$ is a closed subscheme contained in $Z$ and $Z'$ then we get the diagram \begin{diagram} Y & & & &\\ &\rdTo(2,4)\rdTo\rdTo(4,2) & & & \\ & &Z\times_X Z' & \rTo & Z'\\ & &\dTo & & \dTo \\ & &Z & \rTo & X\\ \end{diagram} by the universal property of the fiber product $Z\times_X Z'$, so that $Y$ is contained in $Z\cap Z'$. \begin{proposition} The topological space $|Z\cap Z'|$ of $Z\cap Z'$ is $|Z|\cap |Z'|$. \end{proposition} \begin{proof} Let $U=X-Z\times_X Z'$ and observe that $U$ is open. Observe that $U=(X-Z)\cup (X-Z')$. Indeed, $U,(X-Z),(X-Z')$ are open subschemes of $X$ and $f:Y\rightarrow X$ factors through $U$ if and only if for all $y\in Y$ we have $f(y)\not\in Z\times_X Z'$, that is, if and only if the map $\Spec\kappa(f(y))\rightarrow Y$ fails to factor through $Z$ and $Z'$ (using the universal property of the fiber product). Since $\Spec \kappa(f(y))$ is reduced, such factorization fails if and only if $f(y)\not\in Z$ or $f(y)\not\in Z'$, {\em i.e.} if and only if $f(y)\in (X-Z)\cup (X-Z')$. Thus, $f:Y\rightarrow X$ factors through $U$ if and only if it factors through $(X-Z)\cup (X-Z')$, so the two opens are uniquely isomorphic as they have the same universal property. \end{proof} We now ask: what is the ideal sheaf $\calI_{Z\cap Z'}\subseteq \O_X$? \begin{proposition} We have $\calI_{Z\cap Z'}=\calI_{Z}+\calI_{Z'}$ inside $\O_X$, where for any two sheaves $\calF,\calG$ on a space $Y$ the sum $\calF+\calG$ is the sheaf image of $\calF\oplus\calG$ under the addition map. \end{proposition} \begin{proof} Everything is local on $X$, so we may assume $X=\Spec A$ and $Z=\Spec A/I$ and $Z'=\Spec A/I'$. Let us compute stalks at $x\in X$. We have $$Z\times_X Z'=\Spec (A/I\otimes_A A/I')=\Spec A/(I+I')$$ and $$\calI_{Z\cap Z'}=(I+I')_{\p}=I_{\p}+I'_{\p}$$ in $A_{\p}=\O_{X,x}$. Thus we have $\calI_{Z\cap Z',x}=\calI_{Z,x}+\calI_{Z',x}=(\calI_{Z}+\calI_{Z'})_x$. \end{proof} As an example, suppose that $\Char k\neq 2$ and put $C_i=\Spec k[x,y]/(y+(-1)^i x^n)$ for $i=1,2$. Then $$C_1\cap C_2=\Spec k[x,y]/(y-x^n,y+x^n)=\Spec k[x]/x^n\hookrightarrow \Spec k[x].$$ \section{Remarks on base change} If $k$ is a field and $X$ a finite type $k$-scheme, it is important to study behavior of properties of morphisms and of $X$ with respect to extension of $k$. So for example, if $k'/k$ is an extension and $X'=X\otimes_k k'=X\times_{\Spec k}\Spec k'$, then $X$ being irreducible does {\em not} imply that $X'$ is, and similarly for reduced. We would like to descend properties. That is, we wish to know that if a property holds after base change, then it held before base change. It will turn out that if $\pi:S'\rightarrow S$ is any base change that is {\em faithfully flat} (i.e. flat and surjective) then most ``nice'' properties will hold after base change if and only if they held before base change. A useful principle for finite type schemes over a field $k$ is that we can check many properties over any algebraically closed extension of a field $k$. We will see this later. Flat maps will turn out to have especially nice properties. For example, let $f:X\rightarrow Y$ be a flat map of locally noetherian schemes and suppose $f(x)=y$. Then $\dim \O_{X,x}=\dim \O_{Y,y}+\dim \O_{X_y,x}$. The key point here is that $\O_{X_y,x}\simeq \O_{X,x}/\m_y\O_{X,x}$. \section{Group schemes} Let $\calC$ be the category of sets and $\{\cdot\}$ the final object. A {\em group object} $G$ is an object $G$ together with a triple of morphisms: $$\left(G,G\times G\stackrel{m}{\rightarrow} G,G\stackrel{i}{\rightarrow} G,\{\cdot\}\stackrel{e}{\rightarrow} G\right)$$ such that the following diagrams commute: \begin{enumerate} \item Associativity: \begin{diagram} G\times G\times G & \rTo^{\id\times m} & G\times G \\ \dTo^{m\times \id} & & \dTo_m\\ G\times G & \rTo_{m} & G\\ \end{diagram} \item Identity: \begin{diagram} G\times\{\cdot\}=\{\cdot\}\times G& \rTo^{e\times \id} & G\times G \\ \dTo^{\id\times e} &\rdTo_{\id} & \dTo_m\\ G\times G & \rTo_{m} & G\\ \end{diagram} \item Inverse: \begin{diagram} G & \rTo^{\id\times i} & & &G\times G \\ \dTo^{i\times \id} &\rdTo & & &\dTo_m \\ & &\{\cdot\} & & \\ & & &\rdTo^{e} & \\ G\times G & \rTo_{m} & & &G \\ \end{diagram} \end{enumerate} It is clear that in any category with finite products and a final object we have a notion of a group object. For example, let $\calC$ be the category of $S$-schemes. Here the final object is $S$ and products are fiber products over $S$. So an $S$-group $G$ is a scheme $\pi:G\rightarrow S$ together with a section $e:S\rightarrow G$ such that $\pi\circ e=e\circ \pi=\id$ and maps $m:G\times_S G\rightarrow G$ and $i:G\rightarrow G$ making the above three diagrams commute. As an example, consider $\G_a=\A^1_{\Z}$. We have $\G_a(Y)=\Hom(Y,\A_{\Z}^1)=\Gamma(Y,\O_Y)$, which is an additive group, and this is natural in $Y$, {\em i.e.} a map $Y'\rightarrow Y$ gives a map of groups $\G_a(Y)\rightarrow \G_a(Y')$. It will follow by Yoneda's lemma, which we will see later, that $\G_a$ is a group scheme. Another important example is $\GL_n=\Spec(\Z[X_i]_(\det))$, and $\GL_n(Y)$ is the set of $n\times n$ matrices having entries in $\Gamma(Y,\O_Y)$ with determinant in $\Gamma(Y,\O_Y^{\times})$, functorially in $Y$ as a group via multiplication. The simplest case is $\G_m=\GL_1=\Spec \A^1_{\Z}-\{0\}=\Spec \Z[X,X^{-1}]$. Here, $\G_m(Y)=\Gamma(Y,\O_Y^{\times})$. The map $\G_m(Y)\rightarrow \G_m(Y)$ given by $u\mapsto u^n$ has kernel $\mu_n(Y)$, the $Y$-points of $\mu_n=\Spec \Z[X]/(X^n-1)$. This is a closed subscheme of $\G_m$. We can even consider the $\mu_p/\F_p$, that is, the fiber of $\mu_p\rightarrow \Spec \Z$ over $p$. This is naturally a group scheme, $\Spec \F_p[T]/(T^p-1)$ whose $Y$ points are $\mu_p(Y)=\{u\in \Gamma(Y,\O_Y^{\times}): u^p=1\}$. \end{document}