Fix error in statement and proof of Tag 04VA

Thanks to wu.x
http://stacks.math.columbia.edu/tag/000H#comment-316

Author: aisejohan

sets.tex

@@ -626,31 +626,54 @@ \section{Constructing categories of schemes}
 In
 Algebra, Lemma \ref{algebra-lemma-epimorphism-cardinality}
 we will show that if $A \to B$ is an epimorphism of rings, then
-$|B| \leq |A|$. The analogue for schemes is the following lemma.
+$|B| \leq \max(|A|, \aleph_0)$.
+The analogue for schemes is the following lemma.
 
 \begin{lemma}
 \label{lemma-bound-monomorphism}
-If $X \to Y$ is monomorphism of schemes, then
-$\text{size}(X) \leq \text{size}(Y)$.
+Let $f : X \to Y$ be a monomorphism of schemes.
+If at least one of the following properties
+holds, then $\text{size}(X) \leq \text{size}(Y)$:
+\begin{enumerate}
+\item $f$ is quasi-compact,
+\item $f$ is locally of finite type,
+\item add more here as needed.
+\end{enumerate}
+However, the bound does not hold for general monomorphisms.
 \end{lemma}
 
 \begin{proof}
 Let $Y = \bigcup_{j \in J} V_j$ be an affine open covering of $Y$
-with $|J| \leq \text{size}(Y)$. By
-Lemma \ref{lemma-bound-size}
+with $|J| \leq \text{size}(Y)$. By Lemma \ref{lemma-bound-size}
 it suffices to bound the size of the inverse image of $V_j$ in $X$.
-Hence we reduce to the case that $Y$ is affine.
-As $X \to Y$ is a monomorphism the map $X \to Y$ is injective on underlying
-sets of points. For each $x \in X$ choose an affine open neighbourhood
-$U_x \subset X$. Then $U_x \to Y$ is a monomorphism too, and
-$X = \bigcup_{x \in X} U_x$ is an affine open covering whose index set
-has cardinality at most $\text{size}(Y)$. Hence applying
-Lemma \ref{lemma-bound-size}
-again we see that we reduce to the case that both $X$ and $Y$ are affine.
-In this case the result follows from
-Lemma \ref{lemma-bound-affine}
-and the lemma mentioned just above the statement of the lemma whose
-proof you are reading now.
+Hence we reduce to the case that $Y$ is affine, say $Y = \Spec(B)$.
+For any affine open $\Spec(A) \subset X$ we have
+$|A| \leq \max(|B|, \aleph_0) = \text{size}(Y)$, see remark above
+and Lemma \ref{lemma-bound-affine}. Thus it suffices to show
+that $X$ has at most $\text{size}(Y)$ affine opens. This is clear
+if $X$ is quasi-compact, whence case (1) holds.
+In case (2) the number of isomorphism classes of $B$-algebras $A$
+that can occur is bounded by $\text{size}(B)$, because each
+$A$ is of finite type over $B$, hence isomorphic to an algebra
+$B[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$
+for some $n, m$, and $f_j \in B[x_1, \ldots, x_n]$. However, as
+$X \to Y$ is a monomorphism, there is a unique morphism
+$\Spec(A) \to X$ over $Y = \Spec(B)$ if there is one,
+hence the number of affine
+opens of $X$ is bounded by the number of these isomorphism classes.
+
+\medskip\noindent
+To prove the final statement of the lemma consider the ring
+$B = \prod_{n \in \mathbf{N}} \mathbf{F}_2$ and set $Y = \Spec(B)$.
+For every ultrafilter $\mathcal{U}$ on $\mathbf{N}$ we obtain a maximal
+ideal $\mathfrak m_\mathcal{U}$ with residue field $\mathbf{F}_2$;
+the map $B \to \mathbf{F}_2$ sends the element $(x_n)$ to
+$\lim_\mathcal{U} x_n$. Details omitted.
+The morphism of schemes $X = \coprod_\mathcal{U} \Spec(\mathbf{F}_2) \to Y$
+is a monomorphism as all the points are distinct. However the cardinality
+of the set of affine open subschemes of $X$ is equal to the cardinality
+of the set of ultrafilters on $\mathbf{N}$ which is
+$2^{2^{\aleph_0}}$. We conclude as $|B| = 2^{\aleph_0} < 2^{2^{\aleph_0}}$.
 \end{proof}
 
 \begin{lemma}
@@ -690,7 +713,8 @@ \section{Constructing categories of schemes}
 For example this holds if $T$ can be covered by at most
 $|\mathbf{R}| = 2^{\aleph_0} = \aleph_0^{\aleph_0}$ open affines.
 \item For any $S \in \Ob(\Sch_\alpha)$ and
-any monomorphism $T \to S$, there exists a
+any monomorphism $T \to S$ which is either locally of finite type
+or quasi-compact, there exists a
 $S' \in \Ob(\Sch_\alpha)$ with $S' \cong T$.
 \item Suppose that $T \in \Ob(\Sch_\alpha)$ is
 affine. Write $R = \Gamma(T, \mathcal{O}_T)$.

spaces.tex

@@ -2493,8 +2493,9 @@ \section{Change of big site}
 as in the proof of
 Lemma \ref{lemma-change-big-site}).
 Let $R$ be an object of $(\Sch'/S)_{fppf}$ representing
-$h'_U \times_X h'_U$. Note that $R \to U \times_S U$ is a monomorphism
-of schemes. Hence by
+$h'_U \times_X h'_U$. Note that $R \to U \times_S U$ is a
+monomorphism of schemes which is locally of finite type
+(Lemma \ref{lemma-properties-diagonal}). Hence by
 Sets, Lemma \ref{sets-lemma-what-is-in-it}
 the scheme $R$ is isomorphic to an object of $(\Sch/S)_{fppf}$
 and hence we may (after replacing $R$ by an isomorphic scheme) assume