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\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}