added lec08, thanks Nils!

lec.tex

@@ -24,6 +24,9 @@ Christian Merten\\
 \input{lec03}
 \input{lec04}
 \input{lec05}
+\input{lec06}
+\input{lec07}
+\input{lec08}
 
 \bibliographystyle{alpha}
 \bibliography{refs}

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}
+
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+
+\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}