added lec03

lec03.tex

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+\documentclass{lecture}
+\begin{document}
+	
+	\section{Regular Schemes over Fields}
+	
+	\begin{bem}
+		Coming from differential geometry, we have three notions of the tangent space of a manifold $M$ at a point $x\in M$:
+		\begin{itemize}
+			\item $T_xM=\{\alpha:(-\ve,\ve)\to M\mid\ve>0,\;\alpha(0)=x\}/\text{change of charts}$
+			\item $T_xM= \operatorname{Der}(\mathcal{O}_{M,x},\R)$
+			\item $T_xM=\operatorname{Hom}(\mathfrak{m}_x/\mathfrak{m}_x^2,\R)$
+		\end{itemize}
+	\end{bem}
+	
+	\begin{bem}
+		As a reminder: for a noetherian local ring $(A,\mathfrak{m})$ of dimension $d$, the following are equivalent:
+			\begin{itemize}
+				\item $\operatorname{gr}_\mathfrak{m}(A)\cong A/\mathfrak{m}[T_1,\dots,T_d]$,
+				\item $\dim_{A/\mathfrak{m}}(\mathfrak{m}/\mathfrak{m}^2)=d$,
+				\item $\mathfrak{m}$ has a generator set of $d$ elements.
+			\end{itemize}
+		In this case, $A$ is called \emph{regular}.
+		
+		$A$ regular ring will always be an integral domain.
+	\end{bem}
+	
+	\begin{definition}
+		A locally noetherian scheme $X$ is called \emph{regular in $x\in X$}, if $\mathcal{O}_{X,x}$ is a regular noetherian local ring. Write 
+			\[ X_\text{reg}:=\{x\in X\mid X \text{ is regular in x}\}\,. \]
+		We call $X$ regular, if $X_\text{reg}=X$.
+		
+		The \emph{tangent space} of $X$ in $x$ is defined via 
+			\[T_xM:=\operatorname{Hom}_{\kappa(x)}(\mathfrak{m}_x/\mathfrak{m}_x^2,\kappa(x))\,.\]
+	\end{definition}
+
+	\begin{bem*}
+		If $X$ is integral, then $\mathfrak{m}_\eta=0$ and thus $T_\eta X=0$.
+	\end{bem*}
+
+	\begin{bsp}
+		Let $k$ be a field and $f_1,\dots,f_r\in k[T_1,\dots,T_n]$ polynomials. Set $X=V(f_1,\dots,f_r)\subseteq \mathbb{A}^n_k$. For $x\in\mathbb{A}^n_k(k)$ we have an isomorphism
+			\[k^n\to T_x\mathbb{A}^n_k, \quad (v_1,\dots,v_n)\mapsto(\overline{g}\mapsto \sum_i v_i\frac{\partial g}{\partial T_i}(x)) \,. \]
+		The map $k[S_1,\dots,S_r]\to k[T-1,\dots,T_n],\,S_i\mapsto T_i$ induces morphisms $f:\mathbb{A}^n_k\to\mathbb{A}^r_k$ and $df_x:T_x\mathbb{A}^n_k\to T_{f(x)}\mathbb{A}^r_k$ which fits into the following diagram
+			\[\begin{tikzcd}
+				T_x\mathbb{A}^n_k \ar[d,"\cong"]\ar[r, "df_x"] & T_{f(x)}\mathbb{A}^r_k \ar[d,"\cong"]\\
+				k^n \ar[r, "\cdot J(f)"] & k^r.
+			\end{tikzcd}\]
+		Here $J(f)$ denotes the Jacobian. Claim: $T_xX=\ker(df_x)$.
+	\end{bsp}
+	
+	\begin{definition}
+		Set $k[\ve]=k[X]/(X^2)$. For $X/k$ and $x\in X(k)$ define $X(k[\ve])_x$ as the pullback
+			\[\begin{tikzcd}
+				X(k[\ve])_x \ar[r,hook] \ar[d] & X(k[\ve]) \ar[d] \\
+				\{x\} \ar[r, hook]&  X(k).
+			\end{tikzcd}\]
+	\end{definition}
+	
+	\begin{satz}[]
+		We have a bijection $X(k[\ve])_x\xrightarrow{\cong}T_xX$ which is functorial in $(X,x)$.
+		\label{prop:tangspace-as-pullback}
+	\end{satz}
+	
+	\begin{proof}
+		Left as an exercise.			
+	\end{proof}
+	
+	Grothendieck preaches relativity in all things, hence the following definition.
+	\begin{definition}
+		Let $f:X\to Y$ be a morphism of schemes and $d\geq0$. We call $f$ \emph{smooth of relative degree $d$ in $x\in X$}, if there exist neighbourhoods $x\in U\subseteq X$ open, $f(x)\in\Spec(R)=V\subseteq Y$ open affine as well as an $n\geq0$ and polynomials $f_1,\dots,f_{n-d}\in R[T_1,\dots,T_n]$ such that
+			\[\begin{tikzcd}
+				U \ar[rd, "f"'] \ar[r,hook,"\text{open}"] & \Spec(R[T_1,\dots,T_n]/(f_1,\dots,f_{n-d})) \ar[d]\\
+				& V
+			\end{tikzcd}\]
+		commutes and $J_{f_1,\dots,f_{n-d}}(f)\in M_{n-d,n}(\kappa(x))$ is of full rank.
+		
+		Call $f$ \emph{smooth of relative degree $d$}, if this is the case everywhere.
+	\end{definition}
+	
+	\begin{satz}[\cite{gw},6.15]
+		\begin{enumerate}
+			\item If $f:X\to Y$ is smooth in $x\in X$, then $f$ is smooth in an open neighbourhood of $x$.
+			\item Smoothness of relative dimension $d$ is local on source and target. It is closed under base change and composition (where in the latter degree is additive).
+			\item Open immersions are smooth of rel. dimension 0.
+			\item If $f\circ g$ is smooth and $g$ is unramified, then $f$ is smooth.
+		\end{enumerate}
+		\label{prop:smooth-properties}
+	\end{satz}
+
+	\begin{bem}[Relation to étale morphisms] \phantom{text}
+		\begin{itemize}
+			\item étale $\Leftrightarrow$ flat, unramified and locally of finite presentation $\Leftrightarrow$ smooth of rel. dim. 0.
+			\item Let $f:X\to Y$ be of locally finite presentation. Then $f$ is smooth of rel dim. $d$ in $x\in X$ if there exists a commutative diagram
+				\[\begin{tikzcd}[column sep=small]
+					x\in U \ar[rr, "\text{étale}"] \ar[rd,"f"'] &&\mathbb{A}^d_V \ar[ld] \\
+					& f(x)\in V. &
+				\end{tikzcd}\]
+		\end{itemize}
+	\end{bem}
+
+	\begin{bsp}
+		Let $S$ be a schmeme.
+		\begin{itemize}
+			\item The canonical morphisms $\mathbb{A}^n_S\to S$ and $\mathbb{P}^n_S\to S$ are smooth of rel. dim. $n$.
+			\item $S=\Spec(k),\,k\subseteq\overline{k},\,\operatorname{char}(k)\neq2,\,f\in k[T],\,X=V(U^2-f(T))\subseteq\mathbb{A}^2_k=\Spec(K[T,U])$. Then $X$ is smooth iff $f$ is separable.
+			\item $X=\Spec(\Z_p[U,V]/(U^2-V^3-p))$ is regular, but $X\to\Spec(\Z_p)$ is not smooth.
+		\end{itemize}
+	\end{bsp}
+	
+	\begin{lemma}
+		Let $X,Y$ be $k$-schemes and locally of finite type. Let $x\in X,y\in Y$ be points and $\phi:\mathcal{O}_{X,x}\xrightarrow{\cong}\mathcal{O}_{Y,y}$ an isomorphism of $k$-algebras.
+		
+		Then there exist open neighbourhoods $x\in U \subseteq X,\,y\in V\subseteq Y$ and an isomorphism $f:U\xrightarrow{\cong}V$ such that $f(x)=y$ and $f^\#_x=\phi^{-1}$.
+		\label{lem:extnd-iso-of-stlks}
+	\end{lemma}
+	
+	\begin{satz}
+		Let $X/k$ be an integral scheme of finite type and dimension d, and let $K(X)/k$ be separable (to see what this is supposed to mean, have a look at the proof).
+		
+		Then there exists an open and dense subset $U\subseteq X$ and an isomorphism
+			\[ U\cong \Spec(k[T_1,\dots,T_d,T]/(g)) \]
+		where $g\in k(T_1,\dots,T_d)[T]$ is a separable irreducible monic polynomial with coefficients in $k[T_1,\dots,T_d]$.
+		\label{prop:affine-dense-subset-of-integral}
+	\end{satz}
+
+	\begin{proof}
+		Find $T_1,\dots,T_d\in K(X)$ algebraically independent and such that
+			\[ k \hookrightarrow L:=k(T_1,\dots,T_d) \xhookrightarrow[]{\text{alg.\& sep.}}K(X) \]
+		Write $K(X)=L(\alpha)$ and let $g$ be the minimal polynomial of $\alpha$ over $L$. After suitable multliplications, we can assume $g\in k[T_1,\dots,T_d][T]$. Then 
+			\[ \mathcal{O}_{X,\eta} =K(X) \cong K(k[T-1,\dots,T_d][T]/(g)) =\mathcal{O}_{Y,(0)} \]
+		and the proposition follows from Lemma \ref{lem:extnd-iso-of-stlks}.
+	\end{proof}
+
+	\begin{satz}
+		Let $\emptyset\neq X$ be geometrically reducible and locally of finite type over $k$.
+		
+		Then $X_{\text{sm}}:=\{x\in X\mid X\to k\text{ is smooth in $x$}\}\subseteq X$ is open and dense.
+		\label{prop:smooth-locus-dense}
+	\end{satz}
+
+	\begin{proof}
+		The openness was stated in Proposition \ref{prop:smooth-properties}. It suffices to show: for any irreducible component $Z$ of $X$ there exists an $\emptyset\neq U=\Spec(A)\subseteq X$ affine and open such that $U\subseteq Z$ and $U_\text{sm}=X_\text{sm}\cap U$ is dense in $U$.
+		
+		$X$ is locally noetherian, therefore $X$ locally has only finitely many irreducible components. Therefore, for $U\subset Z$ open the set $U\setminus\bigcup_{Z'\neq Z \text{irred. comp.}}(U\cap Z')$ is open in $X$ and wlog we can assume $X$ to be integral.
+		
+		Using \ref{prop:affine-dense-subset-of-integral} and \ref{prop:char-geom-red}, we can assume $X=\Spec(k[T_1,\dots,T_d,T]/(g))$ with $g$ separable and irreducible. Because $g$ is separable, we have $\frac{\partial g}{\partial T}\neq0$. Since $X$ is reduced, this implies that $X_\text{sm}=\{x\in X\mid \exists i\in\{1,\dots,d,\emptyset\}\frac{\partial g}{\partial T_i}(x)\neq0\}\neq\emptyset$ is non-empty and therefore dense.
+	\end{proof}
+\end{document}