\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 $X_K \to \Spec K$ 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}

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