Spec(Z) is not quasi-compact

A bit of a brain teaser. Thanks to Bhargav Bhatt

examples.tex

@@ -5844,6 +5844,321 @@ \section{Universally submersive but not V covering}
 
 
 
+\section{The spectrum of the integers is not quasi-compact}
+\label{section-canonical}
+
+\noindent
+Of course the title of this section doesn't refer to
+the spectrum of the integers as a topological space,
+because any spectrum is quasi-compact as a topological space
+(Algebra, Lemma \ref{algebra-lemma-quasi-compact}).
+No, it refers to the spectrum of the integers in the
+canonical topology on the category of schemes.
+
+\medskip\noindent
+Let $U$ be a nonprincipal ultrafilter on the set $P$ of prime numbers.
+For a subset $T \subset P$ we denote $T^c = P \setminus T$ the complement.
+For $A \in U$ let $S_A \subset \mathbf{Z}$ be the multiplicative
+subset generated by $p \in A$. Set
+$$
+\mathbf{Z}_A = S_A^{-1}\mathbf{Z}
+$$
+Observe that $\Spec(\mathbf{Z}_A) = \{(0)\} \cup A^c \subset \Spec(\mathbf{Z})$
+if we think of $P$ as the set of closed points of $\Spec(\mathbf{Z})$.
+If $A, B \in U$, then $A \cap B \in U$ and $A \cup B \in U$
+and we have
+$$
+\mathbf{Z}_{A \cap B} =
+\mathbf{Z}_A \times_{\mathbf{Z}_{A \cup B}} \mathbf{Z}_B
+$$
+(fibre product of rings). In particular, for any
+integer $n$ and elements $A_1, \ldots, A_n \in U$ the morphisms
+$$
+\Spec(\mathbf{Z}_{A_1}) \amalg \ldots \amalg \Spec(\mathbf{Z}_{A_n})
+\longrightarrow \Spec(\mathbf{Z})
+$$
+factors through $\Spec(\mathbf{Z}[1/p])$ for some $p$
+(namely for any $p \in A_1 \cap \ldots \cap A_n$).
+We conclude that the family of flat morphisms
+$\{\Spec(\mathbf{Z}_A) \to \Spec(\mathbf{Z})\}_{A \in U}$ is jointly
+surjective, but no finite subset is.
+
+\medskip\noindent
+For a $\mathbf{Z}$-module $M$ we set
+$$
+M_A = S_A^{-1}M = M \otimes_{\mathbf{Z}} \mathbf{Z}_A
+$$
+Claim I: for every $\mathbf{Z}$-module $M$ we have
+$$
+M =
+\text{Equalizer}\left(
+\xymatrix{
+\prod\nolimits_{A \in U} M_A \ar@<1ex>[r] \ar@<-1ex>[r] &
+\prod\nolimits_{A, B \in U} M_{A \cup B}
+}
+\right)
+$$
+First, assume $M$ is torsion free. Then $M_A \subset M_P$ for all $A \in U$.
+Hence we see that we have to prove
+$$
+M = \bigcap\nolimits_{A \in U} M_A\text{ inside }M_P = M \otimes \mathbf{Q}
+$$
+Namely, since $U$ is nonprincipal, for any prime $p$ we have
+$\{p\}^c \in U$. Also, $M_{\{p\}^c} = M_{(p)}$ is equal to the
+localization at the prime $(p)$.
+Thus the above is clear because already $M_{(2)} \cap M_{(3)} = M$.
+Next, assume $M$ is torsion. Then we have
+$$
+M = \bigoplus\nolimits_{p \in P} M[p^\infty]
+$$
+and correspondingly we have
+$$
+M_A = \bigoplus\nolimits_{p \not \in A} M[p^\infty]
+$$
+because we are localizing at the primes in $A$. Suppose that
+$(x_A) \in \prod M_A$ is in the equalizer. Denote
+$x_p = x_{\{p\}^c} \in M[p^{\infty}]$. Then the equalizer
+property says
+$$
+x_A = (x_p)_{p \not \in A}
+$$
+and in particular it says that $x_p$ is zero for all but a finite
+number of $p \not \in A$. To finish the proof in the torsion case
+it suffices to show that $x_p$ is zero for all but a finite number
+of primes $p$. If not write $\{p \in P \mid x_p \not = 0\} = T \amalg T'$
+as the disjoint union of two infinite sets. Then either
+$T \not \in U$ or $T' \not \in U$ because $U$ is an ultrafilter
+(namely if both $T, T'$ are in $U$ then $U$ contains $T \cap T' = \emptyset$
+which is not allowed). Say $T \not \in U$. Then $T = A^c$ and this
+contradicts the finiteness mentioned above.
+Finally, suppose that $M$ is a general module. Then we look
+at the short exact sequence
+$$
+0 \to M_{tors} \to M \to M/M_{tors} \to 0
+$$
+and we look at the following large diagram
+$$
+\xymatrix{
+M_{tors} \ar[r] \ar[d] &
+\prod\nolimits_{A \in U} M_{tors, A} \ar@<1ex>[r] \ar@<-1ex>[r] \ar[d] &
+\prod\nolimits_{A, B \in U} M_{tors, A \cup B} \ar[d] \\
+M \ar[r] \ar[d] &
+\prod\nolimits_{A \in U} M_A \ar@<1ex>[r] \ar@<-1ex>[r] \ar[d] &
+\prod\nolimits_{A, B \in U} M_{A \cup B} \ar[d] \\
+M/M_{tors} \ar[r] &
+\prod\nolimits_{A \in U} (M/M_{tors})_A \ar@<1ex>[r] \ar@<-1ex>[r] &
+\prod\nolimits_{A, B \in U} (M/M_{tors})_{A \cup B} \\
+}
+$$
+Doing a diagram chase using exactness of the columns and the result
+for the torsion module $M_{tors}$ and the torsion free module
+$M/M_{tors}$ proving Claim I for $M$.
+This gives an example of the phenomenon in the following lemma.
+
+\begin{lemma}
+\label{lemma-non-fpqc-descent}
+There exists a ring $A$ and an infinite family of flat ring maps
+$\{A \to A_i\}_{i \in I}$ such that for every $A$-module $M$ 
+$$
+M =
+\text{Equalizer}\left(
+\xymatrix{
+\prod\nolimits_{i \in I} M \otimes_A A_i \ar@<1ex>[r] \ar@<-1ex>[r] &
+\prod\nolimits_{i, j \in I} M \otimes_A A_i \otimes_A A_j
+}
+\right)
+$$
+but there is no finite subfamily where the same thing is true.
+\end{lemma}
+
+\begin{proof}
+See discussion above.
+\end{proof}
+
+\noindent
+We continue working with our nonprincipal ultrafilter $U$ on the set $P$
+of prime numners.
+Let $R$ be a ring. Denote $R_A = S_A^{-1}R = R \otimes \mathbf{Z}_A$
+for $A \in U$.
+Claim II: given closed subsets $T_A \subset \Spec(R_A)$, $A \in U$
+such that
+$$
+(\Spec(R_{A \cup B}) \to \Spec(R_A))^{-1}T_A =
+(\Spec(R_{A \cup B}) \to \Spec(R_B))^{-1}T_B
+$$
+for all $A, B \in U$, there is a closed subset $T \subset \Spec(R)$
+with $T_A = (\Spec(R_A) \to \Spec(R))^{-1}(T)$ for all $A \in U$.
+Let $I_A \subset R_A$ for $A \in U$ be the radical ideal cutting out $T_A$.
+Then the glueing condition implies
+$S_{A \cup B}^{-1}I_A = S_{A \cup B}^{-1}I_B$
+in $R_{A \cup B}$ for all $A, B \in U$ (because localization preserves
+being a radical ideal). Let $I' \subset R$ be the set of elements
+mapping into $I_P \subset R_P = R \otimes \mathbf{Q}$.
+Then we see for $A \in U$ that
+\begin{enumerate}
+\item $I_A \subset I'_A = S_A^{-1}I'$, and
+\item $M_A = I'_A/I_A$ is a torsion module.
+\end{enumerate}
+Of course we obtain canonical identifications
+$S_{A \cup B}^{-1}M_A = S_{A \cup B}^{-1}M_B$ for $A, B \in U$.
+Decomposing the torsion modules $M_A$ into their $p$-primary
+components, the reader easily shows that there exist $p$-power torsion
+$R$-modules $M_p$ such that
+$$
+M_A = \bigoplus\nolimits_{p \not \in A} M_p
+$$
+compatible with the canonical identifications given above.
+Setting $M = \bigoplus_{p \in P} M_p$ we find canonical isomorphisms
+$M_A = S_A^{-1}M$ compatible with the above canonical identifications.
+Then we get a canonical map
+$$
+I' \longrightarrow M
+$$
+of $R$-modules wich recovers the map $I_A \to M_A$ for all $A \in U$.
+This is true by all the compatibilities mentioned above and the
+claim proved previously that $M$ is the equalizer of the two
+maps from $\prod_{A \in U} M_A$ to $\prod_{A, B \in U} M_{A \cup B}$.
+Let $I = \Ker(I' \to M)$. Then $I$ is an ideal and
+$T = V(I)$ is a closed subset which
+recovers the closed subsets $T_A$ for all $A \in U$. This proves Claim II.
+
+\begin{lemma}
+\label{lemma-Z-not-quasi-compact}
+The scheme $\Spec(\mathbf{Z})$ is not quasi-compact
+in the canonical topology on the category of schemes.
+\end{lemma}
+
+\begin{proof}
+With notation as above consider the family of
+morphisms
+$$
+\mathcal{W} = \{\Spec(\mathbf{Z}_A) \to \Spec(\mathbf{Z})\}_{A \in U}
+$$
+By Lemma \ref{lemma-universal-effective-epimorphism-affine}
+below and the two claims proved above
+this is a universally effective epimorphism.
+In any category with fibre products, the universal effective epimorphisms
+give $\mathcal{C}$ the structure of a site (modulo some set theoretical
+issues which are easy to fix) defining the canonical topology.
+Thus $\mathcal{W}$ is a covering for the canonical topology.
+On the other hand, we have seen above that any finite subfamily
+$$
+\{\Spec(\mathbf{Z}_{A_i}) \to \Spec(\mathbf{Z})\}_{i = 1, \ldots, n},\quad
+n \in \mathbf{N}, A_1, \ldots, A_n \in U
+$$
+factors through $\Spec(\mathbf{Z}[1/p])$ for some $p$.
+Hence this finite family cannot be a universally effective epimorphism
+and more generally no universally effective epimorphism can refine
+$\{\Spec(\mathbf{Z}_{A_i}) \to \Spec(\mathbf{Z})\}_{i = 1, \ldots, n}$.
+By Sites, Definition \ref{sites-definition-quasi-compact}
+this means that $\Spec(\mathbf{Z})$ is not quasi-compact
+in the canonical topology. To see that our notion of quasi-compactness
+agrees with the topos theoretic definition, see
+Sites, Lemma \ref{sites-lemma-quasi-compact}.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-universal-effective-epimorphism-affine}
+Let $\{f_i : X_i \to X\}_{i \in I}$ be a family of morphisms of affine schemes.
+Assume that for any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$
+we have
+$$
+\Gamma(X, \mathcal{F}) =
+\text{Equalizer}\left(
+\xymatrix{
+\prod\nolimits_{i \in I} \Gamma(X_i, f_i^*\mathcal{F})
+\ar@<1ex>[r] \ar@<-1ex>[r] &
+\prod\nolimits_{i, j \in I}
+\Gamma(X_i \times_X X_j, (f_i \times f_j)^*\mathcal{F})
+}
+\right)
+$$
+Then our family of morphisms is a universal effective epimorphism
+in the category of affine schemes.
+\end{lemma}
+
+\begin{proof}
+Let $T$ be an affine scheme and let $h_i : X_i \to T$ be a family of morphisms
+such that $h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2$ on
+$X_i \times_X X_j$ for all $i, j \in I$. Then
+$$
+\prod h_i^\sharp :
+\Gamma(T, \mathcal{O}_T)
+\to
+\prod \Gamma(X_i, \mathcal{O}_{X_i})
+$$
+maps into the equalizer and we find that we get a ring map
+$\Gamma(T, \mathcal{O}_T) \to \Gamma(X, \mathcal{O}_X)$
+by the assumption of the lemma for $\mathcal{F} = \mathcal{O}_X$.
+This ring map corresponds to a morphism $h : X \to T$ such
+that $h_i = h \circ f_i$. Hence our family is an effective
+epimorphism.
+
+\medskip\noindent
+Let $p : Y \to X$ be a morphism of affines. We will show
+the base changes $g_i : Y_i \to Y$ of $f_i$ form an effective epimorphism
+by applying the result of the previous paragraph.
+Namely, if $\mathcal{G}$ is a quasi-coherent $\mathcal{O}_Y$-module, then
+$$
+\Gamma(Y, \mathcal{G}) = \Gamma(X, p_*\mathcal{G}),\quad
+\Gamma(Y_i, g_i^*\mathcal{G}) = \Gamma(X, f_i^*p_*\mathcal{G}),
+$$
+and
+$$
+\Gamma(Y_i \times_Y Y_j, (g_i \times g_j)^*\mathcal{G}) =
+\Gamma(X, (f_i \times f_j)^*p_*\mathcal{G})
+$$
+by the trivial base change formula
+(Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change}).
+Thus we see the property of the lemma holds for the family $g_i$.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-universal-effective-epimorphism}
+Let $\{f_i : X_i \to X\}_{i \in I}$ be a family of morphisms of affine
+schemes. Assume the assumption of
+Lemma \ref{lemma-universal-effective-epimorphism-affine} holds and that moreover
+for any morphism of affines $Y \to X$ the map
+$$
+\coprod X_i \times_X Y \longrightarrow Y
+$$
+is a quotient map of topological spaces.
+Then our family of morphisms is a universal effective epimorphism
+in the category of schemes.
+\end{lemma}
+
+\begin{proof}
+To check that a family of morphisms of schemes with fixed target
+is an effective epimorphism, it suffices to work locally on the target.
+Thus it suffices to base change our family of morphisms
+by $Y \to X$ with $Y$ affine. Set $Y_i = X_i \times_X Y$.
+Let $T$ be a scheme and let $h_i : Y_i \to Y$ be a family of morphisms
+such that $h_i \circ \text{pr}_1 = h_j \circ \text{pr}_2$
+on $Y_i \times_Y Y_j$. Note that $Y$ as a set is the coequalizer
+of the two maps from $\coprod Y_i \times_Y Y_j$ to $\coprod Y_i$:
+this already follows from the first condition of the lemma
+when applied to skyscraper sheaves at points of $Y$; details omitted.
+Hence there is a set map of underlying sets $h : Y \to T$
+compatible with the maps $h_i$. By the second condition of
+the lemma we see that $h$ is continuous!
+Thus if $y \in Y$ and $U \subset T$ is an affine open
+neighbourhood of $h(y)$, then we can find an affine open
+$V \subset Y$ such that $h(V) \subset U$.
+Setting $V_i = Y_i \times_Y V = X_i \times_X V$
+we can use the result proved in
+Lemma \ref{lemma-universal-effective-epimorphism-affine}
+to show that $h|_V : V \to U \subset T$ comes from a unique
+morphism of affine schemes $h_V : V \to U$ agreeing with $h_i|_{V_i}$
+as morphisms of schemes for all $i$. Gleuing these $h_V$ gives a morphism
+$Y \to T$ as desired. Some details omitted; please compare with
+the proof of
+Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}.
+\end{proof}
+
+
+
+
+
 
 
 \input{chapters}