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lec07.tex

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+\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}