typo

lec03.tex

@@ -25,9 +25,9 @@
 	\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 
+		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 $X_\text{reg}=X$.
+		We call $X$ \emph{regular} if $X_\text{reg}=X$.
 		
 		The \emph{tangent space} of $X$ in $x$ is defined via 
 			\[T_xM:=\operatorname{Hom}_{\kappa(x)}(\mathfrak{m}_x/\mathfrak{m}_x^2,\kappa(x))\,.\]
@@ -40,7 +40,7 @@
 	\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
+		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.
@@ -65,19 +65,18 @@
 		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
+		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.
+		Call $f$ \emph{smooth of relative degree $d$} if this is the case everywhere.
 	\end{definition}
 	
-	\begin{satz}[\cite{gw},6.15]
+	\begin{satz}[\cite{gw},6.15] \phantom{text}
 		\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).
@@ -99,7 +98,7 @@
 	\end{bem}
 
 	\begin{bsp}
-		Let $S$ be a schmeme.
+		Let $S$ be a scheme.
 		\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.