add lecture 2

ws2023/gruppenschemata/lec.tex

@@ -20,5 +20,9 @@ Christian Merten\\
 \tableofcontents
 
 \input{lec01}
+\input{lec02}
+
+\bibliographystyle{alpha}
+\bibliography{refs}
 
 \end{document}

ws2023/gruppenschemata/lec02.tex

@@ -0,0 +1,252 @@
+\documentclass{lecture}
+
+\begin{document}
+
+\section{Useful statements on schemes}
+
+Let $k$ be a field.
+
+\begin{definition}
+    Let $\mathcal{P}$ be a property of schemes over fields. For
+    a $k$-scheme $X$ we say
+    \emph{$X$ is geometrically} $\mathcal{P}$ if for all field extensions
+    $K / k$ the base change $X_K \to \mathrm{Spec}\ K$ is $\mathcal{P}$.
+\end{definition}
+
+\begin{bsp}
+    The $\R$-scheme $X = \mathrm{Spec}\left( \R[x]/(x^2 + 1) \right) $
+    is irreducible but not geometrically irreducible.
+\end{bsp}
+
+\begin{satz}[]
+    For a $k$-scheme $X$ the following are equvialent:
+    \begin{enumerate}[(i)]
+        \item $X$ is geometrically reduced
+        \item for every reduced $k$-scheme $Y$, the fibre product $X \times_k Y$ is reduced.
+        \item $X$ is reduced and for every generic point $\eta \in X$ of an
+            irreducible component of $X$, the field extension
+            $\kappa(\eta) / k$ is separable.
+        \item There exists a perfect field $\Omega$ and an extension $\Omega / k$ such that
+            $X_{\Omega}$ is reduced.
+        \item For all finite and purely inseparable field extensions $K / k$,
+            the base change $X_K$ is reduced.
+    \end{enumerate}
+    \label{prop:char-geom-red}
+\end{satz}
+
+\begin{proof}
+    Reducedness is a local property, so without loss of generality $X = \mathrm{Spec}\ A$. Moreover
+    we may assume that $X$ itself is reduced. Let
+    $\left\{ \eta_i \right\}_{i \in I}$ be the set of generic points of irreducible components
+    of $X$. Then we obtain an inclusion
+    \[
+        A \hookrightarrow \prod_{i \in I} \underbrace{\kappa(\eta_i)}_{= S_i^{-1} A}
+    .\] We claim that for any field extension $L / k$ the ring $A \otimes_k L$ is reduced
+    if and only if for all $i \in I$ the ring $\kappa(\eta_i) \otimes_k L$ is reduced.
+    \begin{proof}[proof of the claim]
+        $(\Rightarrow)$: follows since forming the nilradical commutes with localisations.
+        $(\Leftarrow)$: We have
+        \[
+        A \otimes_k L \hookrightarrow \left( \prod_{i \in I}^{} \kappa(\eta_i)  \right)
+        \otimes_k L
+        \hookrightarrow \prod_{i \in I}^{} \kappa(\eta_i) \otimes_k L
+        .\]
+    \end{proof}
+    The claim immediatly implies the equivalence of (iii), (iv), (v) and (1). Since
+    (ii) trivially implies (i). It remains to show that (iii) implies (2).
+    Without loss of generality we may take $Y = \mathrm{Spec}\ B$
+    and set $\{\lambda_j\}_{j \in J}$ to be the generic points of $Y$. Then we obtain
+    \[
+    A \otimes_k B \hookrightarrow
+    A\otimes_k \left( \prod_{j \in J} \kappa(\lambda_j)  \right)
+    \hookrightarrow
+    \left( \prod_{i \in I} \kappa(\eta_i) \right)
+    \otimes_k
+    \left( \prod_{j \in J} \kappa(\lambda_j) \right)
+    \hookrightarrow
+    \prod_{i,j}^{} \underbrace{\kappa(\eta_i) \otimes_k \kappa(\eta_j) }_{\text{reduced}}
+    .\] 
+\end{proof}
+
+\begin{korollar}
+    If $k$ is perfect, then
+    reduced and geometrically reduced are equivalent.
+\end{korollar}
+
+\begin{bem}[]
+    The statements in \ref{prop:char-geom-red} also hold when
+    \emph{reduced} is replaced by \emph{irreducible} or \emph{integral}.
+\end{bem}
+
+\begin{satz}
+    Let $f\colon X \to Y$ be a morphism of schemes that is locally of finite presentation.
+    Then $f$ is open if and only if
+    for every point $x \in X$ and every point $y' \in Y$ with
+    $y = f(x) \in \overline{\{y'\} }$ there exists
+    $x' \in X$ with $x \in \overline{\{x'\} }$ such that $f(x') = y'$.
+    \label{prop:open-stab-gener}
+\end{satz}
+
+\begin{proof}
+    Assume $X = \mathrm{Spec}\ B$ and $Y = \mathrm{Spec}\ A$. 
+    $(\Rightarrow)$: Then set
+    \[
+        Z \coloneqq \mathrm{Spec}\ \mathcal{O}_{X,x}
+        \cap \bigcap_{t \in B \setminus \mathfrak{p}_x} D(t)
+    .\] Since $f$ is open, $y' \in f(D(t))$ for all $t \in B \setminus \mathfrak{p}_x$.
+    Set $f_t \coloneqq f|_{D(t)}$. Then $f_t ^{-1}(y') \neq \emptyset$. For sake
+    of contradiction suppose that $y' \not\in f(Z)$. Then set
+    $g\colon \mathrm{Spec}\ \mathcal{O}_{X,x} \to X \xrightarrow{f} Y$.
+    Therefore
+    \[
+        \emptyset = g^{-1}(y') = \mathrm{Spec}\ \left( \mathcal{O}_{X,x} \otimes_A \kappa(y') \right)
+    .\] Thus
+    \[
+    0 = \mathcal{O}_{X,x} \otimes_A \kappa(y')
+    = \operatorname{colim}_{t \in B \setminus \mathfrak{p}_x}
+    \underbrace{B_t \otimes_A \kappa(y')}_{\neq 0}
+    \] which is a contradiction.
+
+    $(\Leftarrow)$:
+    Show $f(X) \subseteq Y$ is open. By Chevalley's theorem (\cite{gw}, 10.70),
+    the image $f(X)$ is constructible. In the noetherian case
+    use that open is equivalent to constructible and stable under generalizations
+    (\cite{gw}, 10.17). In the general case write $A$ as a colimit of noetherian rings and
+    conclude by careful general nonsense.
+\end{proof}
+
+\begin{lemma}
+    Let $f\colon X \to Y$ be flat, $x \in X$, $y = f(x)$, $y' \in Y$ a
+    generalization of $y$. Then there exists a generalization $x'$ of $x$ such that
+    $f(x') = y'$.
+    \label{lemma:flat-stable-gener}
+\end{lemma}
+
+\begin{proof}
+    Set $A = \mathcal{O}_{Y,y}$, $B = \mathcal{O}_{X,x}$ and
+    $\varphi\colon A \to B$. Since $y \in \text{im}(f)$
+    we have $\mathfrak{m}_yB \neq B$ and
+    $B$ is faithfully flat $A$-module (since $\varphi$ is local and flat). Thus
+    \[
+    0 \neq B \otimes_A \kappa(y')
+    ,\] i.e. $f^{-1}(y') \cap \mathrm{Spec}\ B \neq \emptyset$.
+\end{proof}
+
+\begin{korollar}
+    Let $f\colon X \to Y$ be flat and locally of finite presentation. Then $f$ is universally
+    open.
+\end{korollar}
+
+\begin{proof}
+    From \ref{prop:open-stab-gener} and \ref{lemma:flat-stable-gener} follows
+    that flat and locally of finite presentation implies open. Since the former
+    two properties are stable under base change, the result follows.
+\end{proof}
+
+\begin{korollar}
+    Let $f\colon X \to S$ be locally of finite presentation. If
+    $|S|$ is discrete, then every morphism $X \to S$ is universally open.
+\end{korollar}
+
+\begin{definition}[]
+    Let $f\colon X \to Y$. We say
+    \begin{enumerate}[(i)]
+        \item $f$ is \emph{flat in $x \in X$} if
+            $f_x^{\#}\colon \mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$ is flat.
+        \item $f$ is \emph{flat} if
+            $f$ is flat in every point.
+    \end{enumerate}
+\end{definition}
+
+\begin{bsp}[]
+    \begin{enumerate}[(1)]
+        \item $X \to \mathrm{Spec}\ k$ is flat.
+        \item $\mathbb{A}_{Y}^{n} \to Y$ and
+            $\mathbb{P}_{Y}^{n} \to Y$ are flat.
+        \item Let $f\colon Z \hookrightarrow Y$ be a closed immersion. Then
+            $f$ is flat and locally of finite presentation if and only if $f$ is an open immersion.
+    \end{enumerate}
+\end{bsp}
+
+\begin{satz}
+    The following holds
+    \begin{enumerate}[(i)]
+        \item $\mathrm{Spec}\ B \to \mathrm{Spec}\ A$ is flat if and only if $A \to B$ is flat.
+        \item Flatness is stable under base change and composition.
+        \item Flatness is local on the source and the target.
+        \item Open immersions are flat.
+        \item A morphism $f\colon X \to Y$ is flat if and only if
+            for every $y \in Y$ the canonical morphism
+            \[
+                X \times_Y \mathrm{Spec}(\mathcal{O}_{X,y})
+                \to \mathrm{Spec}(\mathcal{O}_{Y,y})
+            \] is flat.
+    \end{enumerate}
+\end{satz}
+
+\begin{definition}
+    A morphism $f\colon X \to Y$ is called \emph{faithfully flat} if
+    $f$ is flat and surjective.
+\end{definition}
+
+\begin{bsp}[]
+    $\mathrm{Spec}\ \overline{k} \to \mathrm{Spec}\ k$ is faithfully flat.
+\end{bsp}
+
+\begin{lemma}
+    Let $\mathcal{C}$ be a category with equalizers, $F\colon \mathcal{C} \to \mathcal{D}$ a
+    conservative (i.e. reflects isomorphisms) functor that commutes with equalizers. Then
+    $F$ is faithful.
+    \label{lemma:cons-eq-faithful}
+\end{lemma}
+
+\begin{proof}
+    Left as an exercise to the reader.
+\end{proof}
+
+\begin{satz}
+    Is $f\colon X \to Y$ faithfully flat, then
+    $f^{*}\colon \mathrm{QCoh}(Y) \to \mathrm{QCoh}(X)$ faithful.
+    \label{prop:faithfully-flat-faithful-pullback}
+\end{satz}
+
+\begin{proof}
+    Can be deduced from \ref{lemma:cons-eq-faithful}. The details are left to the reader.
+\end{proof}
+
+\begin{bem}[Faithfully flat descent]
+    The statement from \ref{prop:faithfully-flat-faithful-pullback} can be
+    - from a carefully selected viewpoint - viewn as the statement
+    that the functor $X \mapsto \mathrm{QCoh}(X)$ satisfies the sheaf condition
+    for faithfully flat and quasicompact morphisms, i.e. that the diagram
+    \[
+        \begin{tikzcd}
+            \mathrm{QCoh}(Y)
+            \arrow{r}{f^{*}}
+            & \mathrm{QCoh}(X)
+            \arrow[yshift=2pt]{r}{\text{pr}_1^{*}}
+            \arrow[swap, yshift=-2pt]{r}{\text{pr}_2^{*}}
+            &\mathrm{QCoh}(X \times_Y X)
+            \arrow[yshift=4pt]{r}
+            \arrow[yshift=0pt]{r}
+            \arrow[yshift=-4pt]{r}
+            &
+            \underbrace{\mathrm{QCoh}(X \times_Y X \times_Y X)}_{\text{corresponds to the cocycle condition}}
+        \end{tikzcd}
+    \] is a limit diagram.
+\end{bem}
+
+\begin{satz}[\cite{gw}, 14.53]
+    Let $f\colon X \to Y$ be a $S$-morphism and
+    $g\colon S' \to S$ faithfully flat and quasicompact.
+    Denote by $f' = f \times_S S'$. If $f'$ is
+    \begin{enumerate}[(i)]
+        \item (locally) of finite type or (locally) of finite presentation,
+        \item isomorphism / monomorphism,
+        \item open / closed / quasicompact immersion,
+        \item proper / affine / finite,
+    \end{enumerate}
+    then $f$ has the same property.
+\end{satz}
+
+\end{document}

ws2023/gruppenschemata/refs.bib

@@ -0,0 +1,15 @@
+@book {gw,
+    AUTHOR = {G\"{o}rtz, Ulrich and Wedhorn, Torsten},
+     TITLE = {Algebraic geometry {I}},
+    SERIES = {Advanced Lectures in Mathematics},
+      NOTE = {Schemes with examples and exercises},
+ PUBLISHER = {Vieweg + Teubner, Wiesbaden},
+      YEAR = {2010},
+     PAGES = {viii+615},
+      ISBN = {978-3-8348-0676-5},
+   MRCLASS = {14-01},
+  MRNUMBER = {2675155},
+MRREVIEWER = {C\'{\i}cero\ Carvalho},
+       DOI = {10.1007/978-3-8348-9722-0},
+       URL = {https://doi.org/10.1007/978-3-8348-9722-0},
+}