\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}
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