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| \documentclass{lecture} | |||
| \begin{document} | |||
| \begin{lemma} | |||
| Let $X/k$ be of locally finite type. | |||
| Then $X_\text{sm}$ is constructible. | |||
| \end{lemma} | |||
| \begin{proof} | |||
| WLOG $X$ is affine. Then the assertion follows from B53, B72c and OCG13Z. | |||
| \end{proof} | |||
| \begin{satz} | |||
| If $f:X\to Y$ is a morphism of schemes and $|Y|$ is discrete, then $f$ is universally open. (cf. Corollary \ref{cor:lfp+discrete-target->univ-open}) | |||
| \end{satz} | |||
| \begin{proof} | |||
| Universal openness is local on the target, therefore wlog $\#Y=1$. | |||
| Since, in addition, universal openness is a topological condition, we can assume $Y$ to be reduced. Therefore let $Y=\Spec k$ for $k$ a field. | |||
| Let $Y'\to Y$ be arbitrary. Since openness is local on the domain, assume $X=\Spec A$; $Y'=\Spec B$ and therefore $X\times_Y Y'=\Spec A\otimes_kB$. Write $A=\operatorname{colim}_\alpha A_\alpha$ as colimit over the finitely generated subalgebras $A_\alpha\subseteq A$. Then | |||
| \[ A\otimes_kB = \operatorname{colim}_\alpha(A_\alpha\otimes_kB)\,. \] | |||
| Let $t\in B$ and denote by $f'$ the base change of $f$. We show that $U=f'(D(t))$ is open in $\Spec B$. Let $t\in A_\alpha\otimes_k B$ for suitable $\alpha$. Applying Corollary \ref{cor:flat+lfp->univ-open} shows that $f'':\Spec A_\alpha\otimes_kB\to\Spec B$ is open. Therefore $U'=f''(D(t))\subseteq\Spec B$ is open, so it suffices to check $U=U'$. | |||
| We have $U\subseteq U''$ by assumption.Let $y\in U'$. It suffices to show | |||
| \[ (f')^{-1}(y)\cap D(t)\neq\emptyset\,. \] | |||
| But $(f')^{-1}=g^{-1}(W)$ with $g:(f')^{-1}(y)=\Spec(B'\otimes_B\kappa(y)), (f'')^{-1}(y)=\Spec (A_\alpha\otimes_k\kappa(y))$ and $W=(f'')^{-1}(y)\cap D(t)$. Since $\kappa(y)/k$ is flat, we have an injection $A_\alpha\otimes_k\kappa(y)\hookrightarrow A_\alpha\otimes_k\kappa(y)$ and $g$ is dominant. This implies $g^{-1}(W)\neq\emptyset$, since W is open and non empty. | |||
| \end{proof} | |||
| \section{Differentials and Smoothness} | |||
| \begin{definition} | |||
| $A\to B$, $M\in B \operatorname{-Mod}$. Define | |||
| \[ \operatorname{Der}_A(B,M)=\{D\in\\operatorname{Hom}_A(B,M)\mid D(Fg)=fD(g)+gD(f) \quad\forall f,g\in B\} \,.\] | |||
| The module of Kähler differentials of $B/A$ is a pair $(\Omega^1_{B/A},d_{B/A})$ with $\Omega^1_{B/A}\in B\operatorname{-Mod},d_{B/A}\in\operatorname{Der}_A(B,\Omega^1_{B/A})$ such that $d_{B/A,\ast}:\operatorname{Hom}_B(\Omega^1_{B/A},M)\xrightarrow{\cong}\operatorname{Der}_A(B,M)$ is an isomorphism. | |||
| \end{definition} | |||
| \begin{lemma} | |||
| For $A\to B$, we have | |||
| \[\Omega^1_{B/A}\cong \bigoplus_{b\in B} db B / \Big\langle\begin{aligned} | |||
| d(bb')=dbdb'+b'db \\ d(b+b')=db+db'\,, \quad da=0 | |||
| \end{aligned} \mid b,b'\in B, a\in A\Big\rangle\,.\] | |||
| The universal differential $d_{B/A}$ is given by $b\mapsto[db]$. For $I=\operatorname{ker}(B\otimes_AB\to B)$, we have an isomorphism | |||
| \[\Omega^1_{B/A}\to I/I^2\,, \quad [db]\mapsto \overline{1\otimes b-b\otimes 1} \,.\] | |||
| \end{lemma} | |||
| \begin{proof} | |||
| This formal calculation. | |||
| \end{proof} | |||
| \begin{bsp} | |||
| \begin{enumerate} | |||
| \item Let $B=A[T_1,\dots,T_n]$. Then $\Omega^1_{B/A}=\bigoplus_{i=1}^n dT_i B$. | |||
| \item Let $L/K$ be an separable extension. Then $\Omega^1_{L/K}=0$. | |||
| \end{enumerate} | |||
| \end{bsp} | |||
| \begin{lemma} | |||
| Let $A', B$ be $A$-algebras and $S\subseteq A$ multiplicatively closed. Then we have $S^{-1}\Omega^1_{B/A}=\Omega^1{S^{-1}B/A}$ and $\Omega^1{B/A}\otimes_B B' = \Omega^1{B'/A'}$. | |||
| \end{lemma} | |||
| \begin{lemma} | |||
| $f:A\to B, g:B\to C$. Then we have an exact sequence | |||
| \[ \Omega^1_{B/A}\otimes_BC\to \Omega^1_{C/A}\to\Omega^1_{C/B}\to 0 \] | |||
| of $C$-modules. If $g$ is surjective with kernel $I$, the sequence | |||
| \[ I/I^2\to\Omega^1_{B/A}\otimes_BC\to \Omega^1_{C/A}\to 0 \] | |||
| is exact. | |||
| \end{lemma} | |||
| \begin{bsp} | |||
| Let $A$ be a ring and $B=A[T_1,\dots,T_n]/(f_1,\dots,f_n)$. Then $$\Omega^1_{B/A}\cong \bigoplus_{i=1}^ndT_iB/\langle df_i\mid i=1,\dots,n\rangle\,,$$ where $df_i=\sum_k\frac{\partial f_i}{\partial T_k}T_k$. | |||
| \end{bsp} | |||
| \begin{definition} | |||
| Let $i:Y\hookrightarrow X$ be an immersion $Y\xrightarrow{\text{closed}}U\xrightarrow{\text{open}}X$ and $I$ the associated ideal sheaf. Then define $\omega_{Y/X}=I/I^2$ as an $\mathcal{O}_Y$-module. | |||
| For $f:X\to S$ the module of Kähler differentials is given by | |||
| \[\Omega^1_{X/S}:=\omega_{X}/X\times_SX\] | |||
| with $\Delta:X\to X\times_SX$. | |||
| \end{definition} | |||
| \begin{bem} | |||
| $X\to S$ mono implies $\Omega^1_{X/S}=0$. | |||
| \end{bem} | |||
| \begin{satz} | |||
| \begin{enumerate} | |||
| \item For $X\xrightarrow{f}Y\to S$ we have an exact sequence | |||
| \[f^\ast\Omega^1_{Y/S}\to\Omega^1_{X/S}\to\Omega^1_{X/Y}\to 0 \,.\] | |||
| \item Given $X\to S \leftarrow Y$, we have an isomorphism | |||
| \[ p_1^\ast\Omega^1_{X/S}\oplus p_2^\ast\Omega^1_{Y/S}\cong\Omega^1_{X\times_SY} \,. \] | |||
| \item Given $Z\xrightarrow{\text{closed}}X\to S$, we have | |||
| \[\omega_{Z/Y}\to i^\ast \Omega^1_{X/S}\to\Omega^1_{Z/S}\to 0\,.\] | |||
| \item If $X\to S$ is of locally finite type, then $\Omega^1_{X/S}$ is of finite presentation. | |||
| \end{enumerate} | |||
| \end{satz} | |||
| \begin{satz}[\cite{gw}, 18.64] | |||
| Let $k$ be a field, $X/k$ of locally finite type and $x\in X$. | |||
| Then $X$ is smooth in $X$ iff $\Omega^1_{X/S}$ is a free $\mathcal{O}_{X,x}$-module of dimension $\operatorname{dim} X$. | |||
| \end{satz} | |||
| \begin{satz} | |||
| Let $X'=X\times_S\times S'$ be cartesian. Then there exists a canonical isomorphism | |||
| \[ h^\ast \Omega^1_{X/S}\xrightarrow{\cong}\Omega^1_{X/S} \,. \] | |||
| \end{satz} | |||
| \begin{proof} | |||
| exercise sheet numero sei | |||
| \end{proof} | |||
| \begin{satz} | |||
| Let $\pi:G\to S$ be a group scheme with unit $e\in G(S)$. Then there are the following isomorphisms of $\mathcal{O}_G$-modules | |||
| \[ \Omega^1_{G/S}\cong \pi^\ast e^\ast\Omega^1_{G/S}\cong \pi^\ast \underbrace{\omega_{S/G}}_\text{via $e$} \,. \] | |||
| \end{satz} | |||
| \begin{proof} | |||
| First, consider the cartesian diagram | |||
| \[ \begin{tikzcd} | |||
| G\times_S G \ar[r,"m"] \ar[d,"p_i"] & G \ar[d,"\pi"] \\ G \ar[r,"\pi"] & S | |||
| \end{tikzcd}\] | |||
| for $i=1,2$. It yields | |||
| \[ m^\ast \Omega^1_{G/S}\cong\Omega^1_{G\times_SG/G}\cong p_i^\ast \Omega^1_{G/S} \,. \] | |||
| Consider also $i=(e\pi,\operatorname{id}_G):G\to G\times_SG$. Then | |||
| \[ \Omega^1_{G/S}\cong\operatorname{id}_G^\ast \Omega^1_{G/S}=i^\ast p_2^\ast \Omega^1_{G/S} = i^\ast m^\ast \Omega^1_{G/S} = i^\ast p_1^\ast\Omega^1_{G/S} =(e\pi)^\ast\Omega^1_{G/S}=\pi^\ast e^\ast \Omega^1_{G/S}\,.\] | |||
| Secondly, consider the diagram of sections | |||
| \[\begin{tikzcd} | |||
| S \ar[d, "e"] \ar[r, "e"] &G \ar[d,"\Delta"] \\ | |||
| G \ar[u,"\pi", bend left] \ar[r,"i"] & G\times_SG \,, \ar[u,"p_1",bend left] | |||
| \end{tikzcd}\] | |||
| where $i=(e\pi,\operatorname{id}_G)$. We deduce $e^\ast\Omega^1_{G/S}\cong\omega_e$ and $\pi^\ast$ yields $\pi^\ast e^\ast \Omega^1_{G/S}\cong\pi^\ast \omega_e$. | |||
| \end{proof} | |||
| \end{document} | |||