Schemes and Deligne-Mumford Stacks
1.5 Digression: Topological Spaces and 8 -Topoi
1.5.3 Sober Topological Spaces
Our next goal is to describe the essential image of the embedding LocspaãÑTop.
1.5. DIGRESSION: TOPOLOGICAL SPACES AND8-TOPOI 163 Definition 1.5.3.1. Let X be a topological space. A closed subsetK ĎX is said to be irreducible if it is nonempty and cannot be written as a union K´YK` of proper closed subsetsK´, K`ĹK. A pointxPK is said to be ageneric pointofK ifK is the closure of txu. The spaceX is said to be soberif every irreducible closed subset of X has a unique generic point.
Remark 1.5.3.2. LetX be a topological space. For every point xPX, the closure of the set txu is irreducible.
Example 1.5.3.3. Every Hausdorff topological spaceX is sober (the only irreducible closed subsets ofX are singletons).
Proposition 1.5.3.4. Let Λ be a locale. Then the topological space |Λ|is sober.
Proof. LetU be an indecomposable element of Λ. For eachV PΛ, we have U R |Λ|V if and only ifV ďU. In particular,|Λ|U is the largest open subset of|Λ|which does not contain U, and is therefore the complement of the closure tUu. If U1 is another indecomposable element of Λ having the same closure in |Λ|, then for eachV PΛ we have
U R |Λ|V ôU1 R |Λ|V, so that V ďU ôV ďU1 and thereforeU “U1.
Let K be an irreducible closed subset of|Λ|. The above argument shows thatK has at most one generic point. We will complete the proof by showing that there exists a generic point of K. Set S“ tU PΛ :K “ |Λ| ´ |Λ|Uu. SinceK is closed, the set S is nonempty. It follows thatS contains a largest elementU. We will complete the proof by showing thatU is indecomposable, so thatK “ |Λ| ´ |Λ|U is the closure ofU.
Suppose thatTis a finite subset of Λ satisfyingU “Ź
VPT V. We wish to show thatU PT. Suppose otherwise: then, by the maximality ofU, each of the open sets |Λ|V has nontrivial intersection withK. Since K is irreducible, it follows that the intersectionŞ
VPT |Λ|V has nontrivial intersection with K, contradicting our assumption that |Λ|UXK“ H.
Proposition 1.5.3.5. LetX be a topological space. The following conditions are equivalent:
p1q The topological space X is sober.
p2q The unit map u:XÑ |UpXq| is a homeomorphism.
Proof of Proposition 1.5.3.5. The implicationp2q ñ p1q follows immediately from Proposi-tion 1.5.3.4. For the converse, suppose thatp1q is satisfied. Note that an open setU PUpXq is indecomposable if and only if the complementX´U is irreducible. Consequently, we can identify |UpXq| with the collection of irreducible closed subsets ofX. Under this identifica-tion, the map ucarries a pointxPX to the closure oftxu. The assumption thatX is sober
implies that the mapu is bijective. To complete the proof that u is a homeomorphism, it will suffice to note that every open subset U ĎX is the inverse image (under the mapu) of the open subset |UpXq|U ĎUpXq.
Remark 1.5.3.6. Combining Propositions 1.5.3.4 and 1.5.3.5, we conclude that the category Topsob of sober topological spaces is a localization of the category Top of all topological spaces. The inclusion Topsob ãÑTop admits a left adjoint, given by XÞÑ |UpXq|. We refer to the topological space |UpXq| as the soberification of X. The points of|UpXq| can be identified with irreducible closed subsets ofX.
The above arguments show that the adjoint functors Top U //Loc
||
oo
restrict to an equivalence between the categoryTopsob of sober topological spaces and the category Locspa of spatial locales. Topsob »Locspa
Algebraic geometry furnishes plenty of examples of sober topological spaces.
Proposition 1.5.3.7. Let X be a topological space. Then:
paq If X is sober, then any open subset U ĎX is sober.
pbq If X can be written as a union of sober open subsets Uα, then X is sober.
Proof. We first provepaq. LetK ĎU be an irreducible closed subset ofU, and let K be its closure inX. We first claim thatK is irreducible. Suppose otherwise: then we can write K “K´YK` for some proper closed subsets K´, K` ĎK. In this case, we also have K “ pU XK´q Y pU XK`q. The irreducibility of K then implies that K “ U XK´ or K “U XK`. SinceK is dense inK, this contradicts our assumption that eitherK´ and K` are proper subsets ofK.
IfXis irreducible, we conclude thatK has a unique generic pointxPX. Thenxbelongs to the nonempty open subsetU XK “K of K, and therefore belongs toU. It follows that x is a generic point of K in U. We claim that this generic point is unique. To see this, suppose thaty is any other generic point of K inU. ThenyPKĎK, so that the closure of tyuin X is contained in K. However, the closure oftyu inX contains K and therefore containsK. It follows thaty is a generic point of K in X, so that y“x by virtue of our assumption thatX is sober. This completes the proof of paq.
We now prove pbq. Suppose thatX admits a covering by sober open subsetstUαuαPA. Let KĎX be an irreducible closed subset. ThenK is nonempty, so that the intersection Kα“UαXK is nonempty for some αPA. Then we can writeK “KαY pKX pX´Uαqq. Using the irreducibility ofK, we deduce that K “Kα: that is,Kα is dense in K. We now
1.5. DIGRESSION: TOPOLOGICAL SPACES AND8-TOPOI 165 claim thatKαis an irreducible closed subset ofUα. To see this, suppose thatKα “K´YK`
for some closed subsetsK´, K`ĂKα. Then K “K´YK`, so that the irreducibility of K implies that K “ K´ or K`. Since K´ and K` are closed in Uα, we conclude that Kα“UαXK is equal to eitherK´ orK`.
Since Uα is irreducible, the set Kα has a unique generic pointxPUα. Then the closure of txu in X is given by Kα “ K, so that x is a generic point of K. Let y be any other generic point of K. Then y is contained in the nonempty open subset Kα Ď K, so that y PUα. It follows that y is a generic point of Kα in Uα, so that y “x by virtue of our assumption thatUα is sober. This completes the proof ofpbq.
Corollary 1.5.3.8. Let pX,OXq be a nonconnective spectral scheme. Then the topological space X is sober.
Proof. Using Proposition 1.5.3.7 and Corollary 1.1.6.2, we can reduce to the case where pX,OXq »SpecA for some connectiveE8-ringA, so that X is homeomorphic to |SpecR|
for R“π0A. In this case, every closed setK ĎX can be realized as the vanishing locus of a radical idealI ĎR. Moreover, K is irreducible if and only ifI is a prime ideal, in which case I is the unique generic point ofK (when regarded as an element of|SpecR|).
1.5.4 Locales Associated to 8-Topoi