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110.80 The spectrum of the integers is not quasi-compact

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 10.17.8). No, it refers to the spectrum of the integers in the canonical topology on the category of schemes, and the definition of a quasi-compact object in a site (Sites, Definition 7.17.1).

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 \mathop{\mathrm{Spec}}(\mathbf{Z}_ A) = \{ (0)\} \cup A^ c \subset \mathop{\mathrm{Spec}}(\mathbf{Z}) if we think of P as the set of closed points of \mathop{\mathrm{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

\mathop{\mathrm{Spec}}(\mathbf{Z}_{A_1}) \amalg \ldots \amalg \mathop{\mathrm{Spec}}(\mathbf{Z}_{A_ n}) \longrightarrow \mathop{\mathrm{Spec}}(\mathbf{Z})

factors through \mathop{\mathrm{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 \{ \mathop{\mathrm{Spec}}(\mathbf{Z}_ A) \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} _{A \in U} is jointly surjective, but no finite subset is.

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.

Lemma 110.80.1. 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.

Proof. See discussion above. \square

We continue working with our nonprincipal ultrafilter U on the set P of prime numbers. 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 \mathop{\mathrm{Spec}}(R_ A), A \in U such that

(\mathop{\mathrm{Spec}}(R_{A \cup B}) \to \mathop{\mathrm{Spec}}(R_ A))^{-1}T_ A = (\mathop{\mathrm{Spec}}(R_{A \cup B}) \to \mathop{\mathrm{Spec}}(R_ B))^{-1}T_ B

for all A, B \in U, there is a closed subset T \subset \mathop{\mathrm{Spec}}(R) with T_ A = (\mathop{\mathrm{Spec}}(R_ A) \to \mathop{\mathrm{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

  1. I_ A \subset I'_ A = S_ A^{-1}I', and

  2. M_ A = I'_ A/I_ A is a torsion module.

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 which 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 = \mathop{\mathrm{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.

Lemma 110.80.2. The scheme \mathop{\mathrm{Spec}}(\mathbf{Z}) is not quasi-compact in the canonical topology on the category of schemes.

Proof. With notation as above consider the family of morphisms

\mathcal{W} = \{ \mathop{\mathrm{Spec}}(\mathbf{Z}_ A) \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} _{A \in U}

By Descent, Lemma 35.13.5 and the two claims proved above this is a universal 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

\{ \mathop{\mathrm{Spec}}(\mathbf{Z}_{A_ i}) \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} _{i = 1, \ldots , n},\quad n \in \mathbf{N}, A_1, \ldots , A_ n \in U

factors through \mathop{\mathrm{Spec}}(\mathbf{Z}[1/p]) for some p. Hence this finite family cannot be a universal effective epimorphism and more generally no universal effective epimorphism \{ g_ j : T_ j \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} can refine \{ \mathop{\mathrm{Spec}}(\mathbf{Z}_{A_ i}) \to \mathop{\mathrm{Spec}}(\mathbf{Z})\} _{i = 1, \ldots , n}. By Sites, Definition 7.17.1 this means that \mathop{\mathrm{Spec}}(\mathbf{Z}) is not quasi-compact in the canonical topology. To see that our notion of quasi-compactness agrees with the usual topos theoretic definition, see Sites, Lemma 7.17.3. \square


Comments (2)

Comment #3569 by on

Shouldn't the definition of quasi-compact object (Tag 090H) that is being referred to in the title be referenced earlier than the second-to-last sentence? Something like this:

"No, it refers to the spectrum of the integers in the canonical topology on the category of schemes, and the definition of a quasi-compact object in a site (Definition 7.17.1)."


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