added lec08, thanks Nils!

lec.tex

@@ -21,6 +21,12 @@ Christian Merten\\
 
 \input{lec01}
 \input{lec02}
+\input{lec03}
+\input{lec04}
+\input{lec05}
+\input{lec06}
+\input{lec07}
+\input{lec08}
 
 \bibliographystyle{alpha}
 \bibliography{refs}

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.

lec04.tex

@@ -0,0 +1,194 @@
+\documentclass{lecture}
+
+\begin{document}
+
+\section{Group schemes over a field}
+
+Let $k$ be a field and $S = \Spec k$.
+
+\begin{lemma}
+    Let $G$ be a group scheme over $k$. Then $G \to \Spec k$ is separated.
+\end{lemma}
+
+\begin{proof}
+    Let $\pi \colon G \to S$ the structure morphism. Then
+    $\pi$ is separated if and only if $e\colon S \to G$ is a closed immersion. For
+    any $x \in \mathrm{im}(e) \in G$, choose an affine open neighbourhood
+    $x \in U = \Spec A \subseteq G$.
+    Then $\pi|_{U} \circ e = \mathrm{id}_S$, hence the induced map
+    $A \xrightarrow{\Gamma(e)} k$ has a section $\Gamma(\pi|_U)$ and is therefore
+    surjective. Thus $e$ is a closed immersion.
+\end{proof}
+
+\begin{satz}
+    Let $G$ be a group scheme locally of finite type over $k$. Then
+    $G$ is smooth over $k$ if and only if $G$ is geometrically reduced.
+\end{satz}
+
+\begin{proof}
+    The first direction is immediate, since smoothness is invariant under base change and
+    smooth over a field implies reduced.
+    Conversely, for any field extension $\ell / k$ by a prior result
+    $G$ is smooth over $k$ if and only if $G$ is smooth over $\ell$. Thus
+    we may assume $k = \bar k$. By \ref{idk} and \ref{idk}, we obtain
+    $G_{\mathrm{sm}} \neq \emptyset$. By the transitive action
+    of $G(k)$ on $G$, every closed point is smooth. Since
+    \[
+        G_{(0)} = \{ g \in G  \mid \mathrm{dim} \overline{\{g\}} = 0 \} 
+    \] is very dense in $G$ and $G_{\mathrm{sm}} \subseteq G$ is open, the result follows.
+\end{proof}
+
+\begin{lemma}
+    Let $k$ be perfect and $G$ a group scheme locally of finite type over $k$. Then
+    the induced reduced subscheme $G_{\mathrm{red}}$ is a subgroup scheme of $G$.
+\end{lemma}
+
+\begin{proof}
+    Since $(-)_{\mathrm{red}}$ is a functor, we obtain
+    $i\colon G_{\mathrm{red}} \to G_{\mathrm{red}}$ and
+    $e\colon S \to G_{\mathrm{red}}$. By \ref{idk},
+    reduced is equivalent to geometrically reduced since $k$ is perfect. Thus
+    $G_{\mathrm{red}} \times_k G_{\mathrm{red}}$ is reduced and we obtain
+    \[
+    \begin{tikzcd}
+        G x_k G \arrow{r}{m} & G \\
+        G_{\mathrm{red}} \times_k G_{\mathrm{red}} \arrow{u}
+        \arrow[dashed]{r} & G_{\mathrm{red}} \arrow{u}
+    \end{tikzcd}
+    .\] 
+\end{proof}
+
+\begin{korollar}
+    If $k$ is perfect and $G$ a group scheme locally of finite type over $k$. Then
+    $G_{\mathrm{red}}$ is smooth over $k$.
+\end{korollar}
+
+\begin{lemma}
+    Let $G$ be locally of finite type over $k$. Then $G$ is geometrically irreducible
+    if (and only if) $G$ is connected.
+\end{lemma}
+
+\begin{proof}
+    Since $G(k) \neq \emptyset$, we have a morphism
+    $\Spec k \to G$ and $\Spec k$ is geometrically connected. Thus $G$ is geometrically connected.
+    We may therefore assume $k = \bar k$. Since the statement is purely topological, we may
+    further assume that $G$ is reduced and thus smooth over $k$. Hence
+    $G$ is regular by \ref{idk}, in particular for every $g \in G$ the local ring
+    $\mathcal{O}_{G,g}$ is regular and hence an integral domain. Since $G$ is locally noetherian
+    and connected, the claim follows.
+\end{proof}
+
+\begin{definition}
+    An \emph{abelian variety} over $k$ is a connected, geometrically reduced
+    and proper $k$-group scheme.
+\end{definition}
+
+\begin{bem}
+    Abelian varieties are smooth and geometrically integral.
+\end{bem}
+
+\begin{bsp}
+    Elliptic curves are abelian varieties of dimension $1$.
+\end{bsp}
+
+The goal is now to show that abelian varieties are commutative group schemes.
+
+\begin{lemma}
+    Let $X$ be a proper, geometrically connected and geometrically reduced $k$-scheme and
+    $Y$ an affine $k$-scheme. Then every morphism $X \xrightarrow{f} Y$ factors over a
+    $k$-valued point of $Y$.
+    \label{lemma:constant-of-proper-conn-irred-affine}
+\end{lemma}
+
+\begin{proof}
+    By the Liouville theorem for schemes, the global
+    sections of $\mathcal{O}_{X_{\bar k}}$ is $\bar k$. Since
+    $k \to \bar k$ is flat, we obtain
+    \[
+        \Gamma(X, \mathcal{O}_X) \otimes_k \bar k
+        \xlongrightarrow{\simeq} \Gamma(X_{\bar k}, \mathcal{O}_{X_{\bar k}})
+    .\] Since $k \to \bar k$ is even faithfully flat, we obtain
+    $\Gamma(X, \mathcal{O}_X) \simeq k$.
+
+    Choose an embedding $Y \hookrightarrow \mathbb{A}_k^{(I)}$. Then a
+    morphism $f\colon X \to Y$ is equivalent to a morphism
+    $X \xrightarrow{f} Y \hookrightarrow \mathbb{A}_k^{(I)}$, which is equivalent
+    to the datum of a family of $e_i \in \Gamma(X, \mathcal{O}_X)$ which
+    corresponds to a morphism
+    $\Spec k \xrightarrow{e} \mathbb{A}_k^{(I)}$. Thus by construction we obtain
+    a factorisation
+    \[
+    \begin{tikzcd}
+        X \arrow{r}{f} \arrow[dashed]{d} & Y \arrow{r} & \mathbb{A}^{(I)} \\
+        \Spec k \arrow{rru}
+    \end{tikzcd}
+    \] where the dashed arrow is induced from the isomorphism $\Gamma(X, \mathcal{O}_X) \simeq k$.
+\end{proof}
+
+\begin{lemma}[Rigidity]
+    Let $X$ be a geometrically reduced, geometrically connected and proper $k$-scheme
+    with $X(k) \neq \emptyset$. Let further $Y$ be an integral scheme over $k$, $Z$
+    be a separated $k$-scheme and $f\colon X \times_k Y \to Z$ a morphism such that
+    there exists $y \in Y(k)$ such that
+    $f|_{X_{y}}$ factors via a $k$-point $z \in Z(k)$. Then
+    $f$ factors via $\mathrm{pr}_2$.
+    \label{lemma:rigidity}
+\end{lemma}
+
+\begin{proof}
+    Consider the composition
+    \[
+        g\colon X \times_k Y \xrightarrow{pr_2} Y \simeq \Spec k \times_k Y
+        \xrightarrow{(x_0, \mathrm{id})} X \times_k Y \xrightarrow{f} Z
+    \] where $x_0$ is an arbitrarily chosen $k$-rational point of $X$.
+    It remains to show that $f = g$. Choose an open affine
+    neighbourhood $z \in U \subseteq Z$. Then
+    $X_y = \mathrm{pr}_2^{-1}(y) \subseteq f^{-1}(U)$. Since
+    $X$ is proper, $\mathrm{pr}_2$ is a closed map. Thus there
+    exists a $y \in V \subseteq Y$ open
+    with $\mathrm{pr}_2^{-1}(V) \subseteq f^{-1}(U)$. For
+    any $y' \in V$, we obtain
+    \[
+    \begin{tikzcd}
+        X \times_k Y \arrow{r}{f} & Z \\
+        X_{y'} \arrow[dashed, swap]{d}{\alpha(y')}
+        \arrow[hookrightarrow]{u} \arrow[dashed]{r} & U \arrow[hookrightarrow]{u} \\
+        U \times_k \kappa(y') \arrow{ur}
+    \end{tikzcd}
+    .\] By \ref{lemma:constant-of-proper-conn-irred-affine}, the morphism
+    $\alpha(y')$ factors over a $\kappa(y')$-valued point. Thus
+    $f$ and $g$ agree on the dense open subset $X \times_k V$. By reduced-to-separated,
+    the result follows.
+\end{proof}
+
+\begin{korollar}
+    Let $A$ and $B$ be abelian varieties over $k$
+    and $f$ a morphism of $k$-schemes $A \to B$. If under the induced
+    map $f(k)\colon A(k) \to B(k)$ the identity $e_A$ is mapped to $e_B$.
+    \label{cor:av-group-homs}
+\end{korollar}
+
+\begin{proof}
+    Consider the composition
+    \[
+        g\colon A \times_k A \xrightarrow{(f \circ m_A) \times (i_B \circ m_A \circ (f \times f))}
+        B \times_k B
+        \xrightarrow{m_B}
+        B
+    .\] It remains to show that the image of $g$ is precisely $\{e_B\} $. By
+    assumption $f(e_A) = e_B$ and thus
+    \[
+    g(\{e_A\} \times_k A) = \{ e_B\} = g(A \times_k \{e_A\})
+    .\] By repeated application of \ref{lemma:rigidity}, $g$ factors
+    via $\mathrm{pr}_1$ and $\mathrm{pr}_2$. Thus $g$ is constant and $e_B$ is in the image.
+\end{proof}
+
+\begin{korollar}
+    Every abelian variety is commutative.
+\end{korollar}
+
+\begin{proof}
+    Apply \ref{cor:av-group-homs} on $i\colon A \to A$.
+\end{proof}
+
+\end{document}

lec05.tex

@@ -0,0 +1,193 @@
+\documentclass{lecture}
+
+\begin{document}
+
+\begin{lemma}[]
+    Let $X$ be a connected scheme over $k$ and $Y$ a geometrically connected scheme over $k$. If
+    $\mathrm{Hom}_k(Y, X) \neq \emptyset$, then $X$ is geometrically connected.
+\end{lemma}
+
+\begin{proof}
+    Use that $X_{\bar k} \to X$ is an open and closed immersion. Let
+    $\emptyset \neq Z \subseteq X_{\bar k}$ be open and closed. Consider
+    the commutative diagram
+    \[
+    \begin{tikzcd}
+        \bar f^{-1}(Z) = Z \times_k Y \arrow{r} \arrow{d} & Y_{\bar k} \arrow{r} \arrow{d}{\bar f} & Y \arrow{d}{f} \\
+        Z \arrow[hookrightarrow]{r} & X_{\bar k} \arrow{r}{\pi} & X
+    \end{tikzcd}
+    .\] We obtain $\bar f^{-1}(Z) = Y_{\bar k}$. Set $Z' = Y_{\bar k} \setminus Z$. If $Z'$ is
+    not-empty, then by the same argument $\bar f^{-1}(Z') = Y_{\bar k}$. Contradiction.
+\end{proof}
+
+\begin{satz}
+    Let $G$ be a group scheme locally of finite type over $k$.
+    \begin{enumerate}
+        \item If $U, V \subseteq G$ are open and dense. Then $U V = G$ as topological spaces.
+        \item If $G$ is irreducible, then $G$ is quasi-compact.
+        \item Any subgroupscheme $H \subseteq G$ is a closed subscheme.
+    \end{enumerate}
+\end{satz}
+
+\begin{proof}
+    We reduce to $k = \bar k$.
+    \begin{enumerate}[]
+        \item We know that $G_{\bar k} \to G$ is an open and closed immersion. Taking
+            pre-images then preserves open and dense (???) and the result follows.
+        \item By \ref{???} $G$ is geometrically irreducible and $G_{\bar k} \to G$ is surjective, i.e.
+            the quasi-compactness of $G_{\bar k}$ implies the quasi-compactness of $G$.
+        ´\item By \ref{???}, being a closed immersion can be tested by faithfully flat descent.
+    \end{enumerate}
+    Now suppose $k = \bar k$.
+    \begin{enumerate}
+        \item It suffices to show that $U(k) V(k) = G(k)$, since
+            $\overline{U(k)V(k)}$ is very dense in $\overline{UV}$. Since
+            $i\colon G \to G$ is an isomorphism of schemes, $V(k)^{-1} \subseteq G(k)$ is
+            open and dense. Thus
+            for all $g \in G$, $g(V(k)^{-1})$ is open and dense. Thus there
+            exists $u \in g(V(k)^{-1})\cap U(k)$, i.e. there
+            exists $v \in V(k)$ such that $gv^{-1} = u$, i.e. $g = u v$.
+        \item Let $U \subseteq G$ be open, dense and quasi-compact. Then $U \times_k U$ is
+            quasi-compact and $G = \mathrm{im}(U \times_k U \to G)$ is quasi-compact.
+        \item Put the induced reduced subscheme structure on $\bar H \subseteq G$. By
+            \ref{???}, the maps $H \to \Spec k$ and $\bar H \to \Spec k$ are universally open.
+            Since $H \subseteq \bar H$ is dense, we obtain
+            \[
+            H \times_k H \subseteq H \times_k \bar H \subseteq \bar H \times_k \bar H
+            \] is dense. Since
+            $H \times_k H \subseteq m^{-1}(H) \subseteq m^{-1}(\bar H) \hookrightarrow G \times G$,
+            we obtain topologically $\bar H \times \bar H \subseteq m^{-1}(\bar H)$. Since
+            the objects in the lower row are reduced, we therefore obtain a factorisation
+            \[
+                \begin{tikzcd}
+                    G \times G \arrow{r}  & G \\
+                    \bar H \times_{k} \bar H \arrow[hookrightarrow]{u}
+                    \arrow[dashed]{r} & \bar H \arrow[hookrightarrow]{u}
+                \end{tikzcd}
+            .\] Thus $\bar H \subseteq G$ is a subgroupscheme. Thus
+            $H = H \times H = \bar H$ where the last equality follows from 1.
+    \end{enumerate}
+\end{proof}
+
+\begin{definition}
+    Let $G$ be a group scheme locally of finite type over $k$ and $e\colon \Spec k \to G$ is the unit.
+    Then denote by $G^{0}$ the connected component of $G$ that contains $\mathrm{im}(e)$. We call
+    $G^{0}$ the \emph{unit component} of $G$.
+\end{definition}
+
+\begin{bem}
+    Since $G$ is locally noetherian, $G^{0}$ is open and closed.
+\end{bem}
+
+\begin{satz}
+    Let $G$ be a group scheme locally of finite type over $k$.
+    \begin{enumerate}[]
+        \item $G^{0}$ is a quasi-compact, geometrically-irreducible and normal subgroupscheme of $G$.
+        \item Any group morphism $G \to H$ with $H$ locally of finite type over $k$ induces
+            a group homomorphism $G^{0} \to H^{0}$.
+        \item For any field extension $\ell / k$, we have
+            \[
+                (G \times_k \ell)^{0} = G^{0} \times_k \ell
+            .\] 
+    \end{enumerate}
+\end{satz}
+
+\begin{proof}
+   \begin{enumerate}
+       \item Since $G^{0}$ is connected and contains a $k$-rational point, by \ref{???} $G^{0}$ is
+           geometrically connected. Then $G_0 \times_k G_0$ is connected
+           and
+           \[
+           \begin{tikzcd}
+               G \times_k G \arrow{r} & G \\
+               G^{0} \times_k G^{0} \arrow{u} \arrow[dashed]{r} & G^{0} \arrow{u}
+           \end{tikzcd}
+           .\] Since $G^{0} \hookrightarrow G \xrightarrow{i} G$ factors
+           over $G^{0} \hookrightarrow G$, $G^{0}$ is a subgroupscheme.
+           By \ref{???}, $G^{0}$ is geometrically irreducible and therefore
+           by \ref{???} it is quasi-compact.
+           For normality consider a connected component $G'$ of $G$. Then we have a commutative diagram
+           \[
+           \begin{tikzcd}
+               G \times_k G^{0} \arrow{r}{(g, h) \mapsto g h g^{-1}} & G \\
+               G' \times_k G^{0} \arrow[hookrightarrow]{u}
+               \arrow[dashed]{r} & G^{0} \arrow[hookrightarrow]{u}
+           \end{tikzcd}
+           .\] Since $G' \times G^{0}$ is connected, the image of the upper horizontal arrow is
+           in $G^{0}$.
+       \item Any group homomorphism sends the identity to the identity, i.e. the composition
+           $G^{0} \hookrightarrow G \to H$ factors via $H^{0} \hookrightarrow H$.
+        \item Since $G^{0}$ is geometrically connected, the scheme
+            $G^{0} \times_k \ell$ is connected. Moreover
+            $G^{0} \times_k \ell \subseteq G \times_k \ell$ is open and closed. Finally,
+            the identity of $G \times_k \ell$ is contained in $G^{0} \times_k \ell$ by the universal
+            property of the fibre product.
+   \end{enumerate} 
+\end{proof}
+
+The proof of the following lemma is left as an exercise to the reader.
+
+\begin{lemma}
+    Let $G$ be a group scheme locally of finite type over $k$. Then every connected component
+    of $G$ is quasi-compact and geometrically irreducible and $G$ is equidimensional.
+\end{lemma}
+
+\begin{satz}
+    Let $f\colon G \to H$ be a group homomorphism of group schemes locally of finite type over $k$. Then
+    \begin{enumerate}[]
+        \item $\mathrm{im}(f) \subseteq H$ is closed.
+        \item $\mathrm{dim}(G) = \mathrm{dim}(\mathrm{im}(f)) + \mathrm{dim}(\mathrm{ker}(f))$.
+        \item Is $H$ smooth over $k$ and $f$ surjective, then $f$ is faithfully flat.
+    \end{enumerate}
+\end{satz}
+
+\begin{bem}
+    For any integral morphism $f\colon X \to Y$ and $Z \subseteq X$ closed the image
+    $f(Z)$ is closed in $Y$ and $\mathrm{dim}(Z) = \mathrm{dim}(f(Z))$.
+\end{bem}
+
+\begin{proof}
+    Since $H_{\bar k} \xrightarrow{\pi} H$ is integral and surjective
+    and $\mathrm{dim}(Z) = \mathrm{dim}(\pi(Z))$ for any closed subset $Z \subseteq H_{\bar k}$,
+    we may assume $k = \bar k$.
+    \begin{enumerate}
+        \setcounter{enumi}{2}
+        \item Since smooth implies reduced, $H^{0}$ is reduced and by \ref{???} $H^{0}$ is irreducible. Thus
+            $H^{0}$ is integral. By generic flatness, we have a
+            $V \subseteq H^{0}$ that is open and dense such that
+            $f^{-1}(V) \to V$ is flat. Thus for all $h \in H(k)$, the map
+            $f^{-1}(hV) \xrightarrow{f} hV$ is flat. By covering $H$ with
+            translates of $V$, we obtain $f$ is flat.
+        \setcounter{enumi}{0}
+        \item We may assume that $G$ is reduced and thus $G$ is smooth over $k$ by \ref{???}. Let
+            $C$ be $C_{\mathrm{red}} = \overline{f(G)}^{H}$. We claim
+            that $C$ is a subgroupscheme of $H$. Then $G \to C$ is quasi-compact and dominant. Thus
+            we have a factorisation
+            \[
+            \begin{tikzcd}
+                G \times_k G \arrow{r} \arrow{d}{m_G}
+                & C \times_k C \arrow{r} \arrow{d}[dashed]{m_C} & H \times_k H \arrow{d}{m_H} \\
+                G \arrow{r}{f} & C \arrow[hookrightarrow]{r} & H
+            \end{tikzcd}
+            .\] Analogously one obtains
+            \[
+            \begin{tikzcd}
+                C \arrow[hookrightarrow]{d} \arrow[dashed]{r} & C \arrow[hookrightarrow]{d} \\
+                H \arrow{r} & H
+            \end{tikzcd}
+            .\] Thus we may assume that $f$ is dominant.
+            By the theorem of Chevalley, $f(G)$ is constructible and is therefore dense. Hence
+            there exists an open $U \subseteq H$ such that $U \subseteq f(G)$. Thus
+            $H = U \cdot U \subseteq f(G)$ and $f(G) = H$ is closed.
+        \item We may assume that also $H$ is reduced and that $f(G) = H$. Then
+            $H$ is smooth over $k$ and $f$ is flat. By \ref{???} we have $f(G^{0}) \subseteq H$ is open
+            and by 1) also closed. Thus $G^{0} \xrightarrow{f} H^{0}$ is surjective.
+            We have $\mathrm{dim}(G^{0}) = \mathrm{dim}(G)$,
+            $\mathrm{dim}(H^{0}) = \mathrm{dim}(H)$ and
+            $\mathrm{dim}(\mathrm{ker}(f^{0})) = \mathrm{dim}(\mathrm{ker}(f)^{0})$. Now the result follows since
+            all fibres are isomorphic and dimension is additive under flat morphism in non-empty fibres
+            (\cite{gw} 14.119).
+    \end{enumerate}
+\end{proof}
+
+\end{document}

lec06.tex

@@ -0,0 +1,162 @@
+\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}

lec07.tex

@@ -0,0 +1,160 @@
+\documentclass{lecture}
+
+\begin{document}
+	
+	\section{Functor of points}
+	
+	Reference: Demargue-Gabriel: Groups algebraiguexue
+	
+	We want to define quotient group schemes. For scheme $S$, $S$-subgroup $H\hookrightarrow G$ we want a short exact sequence 
+		\[ 0 \to H(T) \to G(T) \to (G/H)(T) \to 0 \,.\]
+	But the presheaf $G/H$ is not generally a sheaf.
+	
+	As an ansatz, consider the yoneda embedding
+		\[ y:\operatorname{Sch}_S \hookrightarrow \operatorname{PSh}(\operatorname{Sch}_S) \,.\]
+	Grothendieck showed: the fpqc-topology is subcanonical, ie. presentable presheaves are sheaves in the fpqc-topology. Therefore, it may be useful to consider the fppf-sheafification of $(T\mapsto G(T)/H(T)).$ (why fppf and not fpqc: later)
+	
+	\begin{bem}Let be: 
+		
+		LRS = category of locally ringed spaces
+		
+		Sch = category of schemes
+		
+		Aff = category of affine schemes
+		
+		All of these are full subcats of each other.
+		
+		For
+			\[ y:\operatorname{Sch}_S \hookrightarrow \operatorname{PSh}(\operatorname{Sch}_S) \,,\]
+		our goal is to consider the essential image of $y$ without reference to $\operatorname{Sch}_S$
+	\end{bem}
+
+\begin{bem}[Ansatz]
+	Schemes are build from affine schemes through glueing on open immersions, i.e. every scheme is a colimit (coequalizer) of affine schemes and open immersions
+		\[ \coprod_{i,j\in I\;k\in J}\Spec(B_{i,j}) \,\substack{\to\\[-1em]\to}\, \coprod_{i\in I}\Spec (A_i) \to X \,.\]
+\end{bem}
+
+\begin{bsp}
+	For scheme $X$ and open subschemes $U,V\subseteq X$ with $U\cup V=X$ the canonical map
+		\[ y(U)\amalg_{y(W)}y(V) \to y(X) \]
+	is not generally an isomorphism in $\operatorname{PSh}(\operatorname{Sch})$ for $W=U\cap V$.
+	
+	But after sheafification $a:\operatorname{PSh}(\operatorname{Sch})\to\operatorname{Sh}(\operatorname{Sch})$ we get two isomorphisms (since presheaves are reflective subcat of sheaves)
+		\[ a(y(U)\amalg_{y(W)}y(V)) \to a(y(U))\amalg_{a(y(W))}a(y(V)) \to a(y(X))=y(X) \,. \]
+\end{bsp}
+
+Question: What is an open immersion between affine schemes?
+
+\begin{satz}
+	Let $f:X\to Y$ be a morphism of schemes. TFAE (?)
+		\begin{enumerate}[(i)]
+			\item $f$ is open immersion,
+			\item $f$ is étale,
+			\item $f$ is flat mono of locally finite presentation.
+		\end{enumerate}
+\end{satz}
+
+\begin{bem}
+	A family $\{\Spec(A_i)\to\Spec(A)\}_{i\in I}$ of open immersions is an {open covering} iff it can be verfeinert by an open covering of the form $\{\Spec(A_{f_\alpha})\to\Spec(A)\}_{\alpha}$.
+	TFAE
+	\begin{enumerate}
+		\item $\{\Spec(A_{f_\alpha})\to\Spec(A)\}_{\alpha}$ is an open covering,
+		\item $A\to \prod_\alpha A_{f_\alpha}$ is faithfully flat,
+		\item $(1)=(f_\alpha\mid\alpha)\subseteq A$.
+	\end{enumerate}
+\end{bem}
+
+$\Sh(\Sch) \simeq \Sh(\Aff)$ and there we only need open immersions:
+
+\begin{bem}
+	Following the preceding remark, we have the sites $\operatorname{Aff}^{\operatorname{Zar}}$ of affine schmes with zariksi topology.
+	
+	The restriction $\operatorname{PSh}(\operatorname{Sch})\to\operatorname{PSh}(\operatorname{Aff})$ induces an equivalence  of cats $\operatorname{Sh}(\operatorname{Sch})\to\operatorname{Sh}(\operatorname{Aff})$ by the comparison lemma.
+	A quasi inverse is given by 
+		\[ F\mapsto  \hat F :X\mapsto \lim_{\Spec A\to X}  F(\Spec A)\,.\]
+	The essential image of
+		\[ \Sch \xhookrightarrow{y} \Sh_{\Zar}(\Sch) \xrightarrow{i^\ast} \Sh_{\Zar}(\Aff) \]
+	consists exactly of those sheaves that can be written as coequalizer of a diagram of the form
+		\[ \coprod_{i,j,k}\Spec(A_{ijk})\,\substack{\to\\[-1em]\to}\,\coprod_i\Spec A_i \]
+\end{bem}
+
+The problem is to check whether something is or is not a sheaf in $\Sh_{\Zar}(\Aff)$.
+
+
+\begin{bem}
+	Let $\Psh$ denote $\Psh(\Aff)=\operatorname{Fun}(\operatorname{CRing},\operatorname{Set})$ and $\Aff=\operatorname{im}(y:\operatorname{CRing}^{\operatorname{op}}\to \operatorname{PSh})$. Set $S(R):=y(R)$ as an "affine scheme".
+	
+	For $A$ ring, $X\in\Psh$, $p\in X(A)$ we have
+		\[ p^\#:S(A)\to X \]
+	in $\Psh$ via $S(A)(R)\to X(R), \phi\mapsto X(\phi)(R)$.
+	
+	A subfunctor $U\hookrightarrow X$ in $\Psh$ is an equivalence class of monos in $\Psh$. 
+\end{bem}
+
+When is a subfunctor an open immersion? "Open immersion" is local on target, so we can check this on open coverings by affine schemes. These are of the shortly following form
+
+\begin{bem}
+	For an ideal $I\subseteq A$ we have decompostion $V(I)\to \Spec (A) \leftarrow D(I)$. For ringmorphism $\phi:A\to R$ have
+		\[ \Spec(\phi)^{-1}(D(I)=D(\phi(I)\cdot R)) \,. \]
+	Therefore
+		\[\begin{tikzcd}
+			\Spec(R) \ar[r] \ar[dr,dashed,"\exists !"] & \Spec(A) \ar[d] \\
+			& D(I)
+		\end{tikzcd}\] 
+	factorizes iff $\phi(I)\cdot R=R$.
+\end{bem}
+
+\begin{definition}
+	\begin{itemize}
+		\item For ideal $I\subseteq A$ define subfunctor $S(A)_I\subseteq S(A)$ via
+			\[S(A)_I(R).=\{\phi:A\to R\mid \phi(I)\cdot R =R\}\]
+			(by the above lemma, these are precisely the points which come from the open subscheme $D(I)$)
+		\item An \emph{open subfunctor} is a subfunctor $U\hookrightarrow X$ such that for every morphism $S(A)\to X$ the projection map from the (pointwise) fibre product $U\times_X S(A)$  to a subfunctor of the form $S(A)_I$ is an isomorphism for a suitable ideal $I\subseteq A$.
+	\end{itemize}
+\end{definition}
+
+\begin{bsp}
+	For $p\neq q$ prime numbers we have $X=\Spec \Z=D(p)\cup D(q)$. But 
+		\[ X(\Z)\supsetneq D(p)(\Z)\cup D(q)(\Z) \,, \]
+	since both sets on the right are empty.
+\end{bsp}
+
+So this isn't quite right either. Maybe fields?
+
+\begin{definition}
+	\begin{itemize}
+		\item A family $(U_i\hookrightarrow X)_{i\in I}$ of open immersions in $\Psh$ is an \emph{open covering}, if for every field $k$ we have
+			\[X(k) = \bigcup_{i\in I} U_i(k) \,.\]
+			(here we could replace "field" by "local ring". The idea is that "points" are specs of fields, and we shouldnt require these covering conditions for all objects, just for points)
+		\item For ring $A$ a \emph{partition of unity} is given by a finite family $(f_i,x_i)$ with $f_i,x_i\in A$ such that
+			\[ \sum_i f_i x_i = 1 \,.\]
+		In this case $(S(A)_{(f_i)})_{i\in I}$ is an open covering of $S(A)$.
+		\item A presheaf $X\in\Psh$ is \emph{local} (ie. it is a \emph{sheaf}) if for all $A\in \operatorname{CRing}$ and all partitions of unity in $A$ the induced diagram
+			\[\begin{tikzcd}
+				X(A) \to \prod_i X(A_{f_i}) \,\substack{\to\\[-1em]\to}\, \prod_{i,j} X(A{f_if_j})
+			\end{tikzcd}\]
+		is a limit-diagram.
+		\item A \emph{scheme} is a local presheaf that allows an open covering by affine schemes.
+	\end{itemize}
+\end{definition}
+
+\begin{bem}
+	An open subfunctor of a scheme is a scheme.
+\end{bem}
+
+\begin{bem}
+	\begin{itemize}
+		\item Let $X=(|X|,\OO_X)$ be an locally ringed space. Obtain $S(X)\in\Psh$ with $S(X)(R)=\Hom_{\LRS}(\Spec(R),X)$
+		\item For $U\subseteq X$ open $S(U)\subseteq S(X)$ open subfunctor.
+		\item A covering $X=\bigcup_{i\in I} U_I$ yields an open covering $(S(U_i))_i$ of $S(X)$.
+	\end{itemize}
+\end{bem}
+
+\begin{satz}
+	A locally ringed space $X$ is a scheme iff $S(X)$ is a scheme.
+\end{satz}
+
+\begin{bem}
+	The Vorschrift $X\mapsto S(X)$ defines a functor $\LRS\to\Psh$ that has a left adjoint.
+\end{bem}
+\end{document}

lec08.tex

@@ -0,0 +1,106 @@
+\documentclass{lecture}
+
+\begin{document}
+		\section*{§7 fppf-topology and algebraic spaces}
+		Let $S$ be a scheme and $Sch_S$ the category of $S$-Schemes. We have the following problem. Given an $S$-group scheme $G$ and an $S$-sub-group-scheme $H$ the quotient $G/H$ may not exist in the category of $S$-schemes. \\
+		We remedy this by forming the quotient in a larger category, namely the category of algberaic spaces or fppf-sheaves, and then study conditions under which the constructed quotient lives in $Sch_S$.
+		\begin{bem}
+			Given a scheme $S$ we have the following topologies on $Sch_S$
+			\[
+			\text{Zariski} \subset \text{étale} \subset \text{fppf} \subset \text{fpqc}.
+			\]
+		\end{bem}
+		\begin{theorem}[Grothendieck, 023Q Stacks]
+			Let $S$ be a scheme. Then the representable $\hom_{Sch}(-,S)$ is a fpqc-sheaf.
+		\end{theorem}
+		\begin{bem}
+			\begin{itemize}
+				\item If $G/H$ as above is a scheme its yoneda image must be an fpqc-sheaf. 
+				\item However sheafification does not work for fpqc-presheaves.
+			\end{itemize}
+		\end{bem}
+		\begin{definition}
+			We call a family of morphisms $\{f_i:X_i\to X\}$ of schemes an fppf-covering iff
+			\begin{enumerate}[1)]
+				\item $f_i$ are flat and of finite presentation $\forall i$ and
+				\item they are jointly surjective, i.e. $X = \bigcup_i f_i(X_i)$.
+			\end{enumerate}
+		\end{definition}
+		\begin{bem}
+			A preseheaf $F:Sch_S^{op} \to Grp$ is an fppf-sheaf iff the associated set-valued presheaf is an fppf-sheaf, since the forgetful functor from groups to sets commutes with limits (as it has an adjoint).
+		\end{bem}
+		Exactness of a sequence of sheaves can be checked just as for topological spaces
+		\begin{bem}
+			Let $\tau$ be a topology on the site $Sch_S$. And let $0\to F\to G\to H$ be a sequence of sheaves with values in abelian groups/modules/\ldots. It is exact iff 
+			\begin{enumerate}[1)]
+				\item for all $X\in Sch_S$ the sequence $0\to F(X)\to G(X) \to H(X)$ is exact and 
+				\item For all $X\in Sch_S$ and all $h\in H(X)$ there is a covering $\{X_i \to X\}$ s.t. $h|_{X_i}$ is in the image of $G(X_i)\to H(X_i)$. 
+			\end{enumerate}
+		\end{bem}
+		\begin{bsp}
+			Let $n\ge 1$ be an integer. The sequence $0\to \mu_{n,S} \to \mathbb{G}_{m,S} \overset{(-)^n}{\to} \mathbb{G}_{m,S}$
+			is an exact sequence of presheaves on $Sch_S$. If $n\in \mathcal{O}_S(S)^\times$, then $(-)^n$ is surjective w.r.t. to the étale topology on $Sch_S$. If $n\notin \mathcal{O}_S(S)^\times$ then $(-)^n$ is not surjective w.r.t. the étale topology but w.r.t. the fppf-topology on $Sch_S$.
+		\end{bsp}
+		Let $\mathcal{C}$ be any category. We denote by $y:\mathcal{C} \to PSh(\mathcal{C}), \ X\mapsto y(X)= \hom_\mathcal{C}(-,X)$ the  yoneda embedding.
+		\begin{definition}
+			\begin{enumerate}[(i)]
+				\item Let $F,G \in PSh(Sch_S)$. We call a morphism $F\to G$ \textit{representable} iff for all $X\in Sch_S$ and all morphisms $y(X)\to G$ the fibre product $F \times_G y(X)$ is representable.
+				\item Let furthermore $\mathbb{P}$ be a property of morphisms of schemes which is closed under pre- and postcomoposition with isomorphisms. We say that a representable morphism $F\to G$ \textit{has $\mathbb{P}$} iff for all $X\in Sch_S$ and all $y(X)\to G$ the morphism of schemes corresponding to 
+				\[
+				F\times_G y(X) \to y(X)
+				\]
+				has the property $\mathbb{P}$.
+			\end{enumerate}  
+		\end{definition}
+		\begin{bem}
+			Note that in the second part of the above definition the object $F\times_G y(X)$ is representable s.t. the definition becomes meaningful.
+		\end{bem}
+		\begin{bem}
+			Representable morphisms are closed under composition and base change. A representable morphism with representable target has representable source, because: Let $F\to G$ be representable, $G=y(X)$. Then for $id:y(X)\to y(X)$ the object $F\times_G G=F$ is representable.
+		\end{bem}
+		\begin{lemma}
+			Let $F\in PSh(Sch_S)$. Then we have 
+			\[
+			\Delta : F\to F \times_S F \text{ representable} \iff \forall X\in Sch_S: \text{every morphism } y(X) \to F \text{ is representable.} 
+			\]
+		\end{lemma}
+		\begin{proof}
+			$\Rightarrow$: Let $y(X) \to F$ and $y(Y)\to F$ be given as in the definition. The diagram 
+			\[
+			\begin{tikzcd}
+				y(X) \times_F y(Y)\arrow{r} \arrow{d} & y(X) \times_S y(Y) = y(X\times_S Y) \arrow{d} \\
+				F \arrow{r}& F\times_S F
+			\end{tikzcd}
+			\]
+			is cartesian. So the upper map is representable with representable target. So the term $y(X)\times_F y(Y)$ is representable.\medskip\\ 
+			$\Leftarrow:$ Let $y(X) \to F\times_S F$. We claim 
+			\[
+			F\times_{F\times_S F} y(X) = y(X) \times_{y(X)\times_S y(X)} (y(X)\times_F y(X)).
+			\]
+			By assumption the term $y(X)\times_F y(X)$ is representable and since then every term on the RHS is representable so is the left term. So it remains to show the claim. For this consider the diagram (we omit the yoneda embedding from the notation)
+			\[
+			\begin{tikzcd}
+				X\times_{X\times_S X} (X\times_F X)\arrow{r}\arrow{d} & X\times_F X \arrow{r}\arrow{d}& F\arrow{d} \\
+				X\arrow{r} & X\times_S X\arrow{r} & F\times_S F.
+			\end{tikzcd}
+			\]
+			Both little square are cartesian. Thus, the outer square is cartesian which shows the claim.
+		\end{proof}
+		\begin{definition}
+			An \textit{algebraic space (over $S)$} is a sheaf $X\in PSh(Sch_S)$ with respect to the fppf-topology s.t. 
+			\begin{enumerate}[i)]
+				\item $X\to X \times_S X$ is representable and 
+				\item There exists an $S$-scheme $U$ and a morphism $y(U) \to X$ which is surjective in the étale topology.
+			\end{enumerate}
+			From this we obtain the full subcategory $AlgSpc_S \subset Sh_{fppf}(Sch_S)$.
+		\end{definition}
+		\begin{bem}
+			The category of algebraice spaces is closed under fibre products in $PSh(Sch_S)$ (what does this mean?)
+		\end{bem}
+		\begin{lemma}
+			Let $Y\in AlgSpc_S$ and $X\to Y$ representable. Then $X\in AlgSpc_S$. 
+		\end{lemma}
+		\begin{proof}
+			The proof of this was not given completely.
+		\end{proof}
+\end{document}