Noetherian spaces and quantifier elimination Matthew de Brecht 1 - - PowerPoint PPT Presentation

Noetherian spaces and quantifier elimination

Matthew de Brecht1

CiNet, NICT, Osaka, Japan

Dagstuhl Seminar: 2016, January 17 - 22

1This work was supported by JSPS Core-to-Core Program, A. Advanced

Research Networks and by JSPS KAKENHI Grant Number 15K15940.

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Introduction

A space is Noetherian iff every strictly ascending chain of

• pen subsets if finite.

Noetherian spaces naturally arise in algebraic geometry as the spectrum of Noetherian rings.

• J. Goubault-Larrecq has also demonstrated that they can be

applied to verification for various classes of transition systems that are not based on well-quasi-orders (wqos).

We prove a quantifier elimination result for countably based sober Noetherian (i.e., quasi-Polish Noetherian) spaces: Assume X and Y are quasi-Polish Noetherian spaces, and let P ⊆ X × Y be constructible. Then ∃XP := {y ∈ Y | (∃x ∈ X)P(x, y)}, and ∀XP := {y ∈ Y | (∀x ∈ X)P(x, y)} are constructible subsets of Y .

This is a weak version of a scheme-theoretic result by Chevellay from algebraic geometry, which is related to quantifier elimination for the theory of algebraically closed fields.

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Table of Contents

1 Noetherian spaces 2 Countably based sober Noetherian spaces

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Table of Contents

1 Noetherian spaces 2 Countably based sober Noetherian spaces

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Noetherian spaces

Definition A topological space is Noetherian iff every open subset is compact. Equivalently, a space is Noetherian iff its open set lattice satisfies the ascending chain condition, iff its lattice of closed sets is well-founded. (Recall that the Alexandroff topology on a quasi-order P is generated by the sets ↑ x for x ∈ P). Thereom (see J. Goubault-Larrecq, 2007) A quasi-order P is a well-quasi-order iff the Alexandroff topology

• n P is Noetherian.

However, the specialization order of a Noetherian space is not always a wqo.

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Noetherian Constructions

• J. Goubault-Larrecq proved that Noetherian spaces are closed

under the following: Subspaces Disjoint unions (X + Y ) The lower power space construction (A(X))

This is the set of closed subsets of X, ordered by inclusion, with the upper topology (the topology generated by complements of sets of the form ↓ x).

Topological products (X × Y )

Proof: X × Y ֒ → A(X) × A(Y ) ∼ = A(X + Y )

Continuous images (in particular, quotients) and several others important constructions... In particular, the category of Noetherian spaces is finitely complete and finitely co-complete, but it is not cartesian closed.

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Sober Noetherian spaces

A space is sober iff every irreducible closed set equals the closure of a single point.

A non-empty closed set is irreducible iff it does not equal the union of two closed proper subsets.

Sober Noetherian spaces are completely characterized by their specialization order: Theorem (J. Goubault-Larrecq, 2007) X is a sober Noetherian space iff The topology on X is the upper topology of a well-founded partial order, and For any finite F ⊆ X there is finite G ⊆ X such that

• x∈F ↓ x =

y∈G ↓ y.

Furthermore, a space is Noetherian iff its soberification is Noetherian, so we don’t lose much by assuming sobriety.

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Table of Contents

1 Noetherian spaces 2 Countably based sober Noetherian spaces

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Preliminaries (1)

The second level of the Borel hierarchy for general spaces (V. Selivanov): A subset A of a topological space X is a Π0

2-set iff there is a

sequence Ui, Vi (i ∈ ω) of open subsets of X such that x ∈ A ⇐ ⇒ (∀i ∈ ω)[x ∈ Ui ⇒ x ∈ Vi] for each x ∈ X.

If X is metrizable then the Π0

2-sets are exactly the Gδ-sets.

The complement of a Π0

2-set is called a Σ0 2-set.

A set which is both Π0

2 and Σ0 2 is called a ∆0 2-set.

We will also use the following terminology (common in algebraic geometry): A subset of a space is constructible if it is equal to a finite boolean combination of open subsets.

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Quantification

Σ0

2-sets correspond to existential quantification over a constructible

predicate with a variable from a countable (discrete) space: If P ⊆ X × Y is constructible (or even Σ0

2) and X is

countable then ∃XP := {y ∈ Y | (∃x ∈ X)P(x, y)} is a Σ0

2-subset of Y .

Conversely, if A is a Σ0

2-subset of Y , then there is a

constructible P ⊆ ω × Y such that A = ∃ωP. Π0

2-sets correspond to universal quantification in a similar sense.

Quantification over more complicated predicates and spaces result in higher levels of the Borel and (hyper-) projective hierarchies.

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Preliminaries (2)

A space X is quasi-Polish iff it is countably based and the topology is generated by a (Smyth-) complete quasi-metric. Equivalently, iff X is homeomorphic to a Π0

2-subset of P(ω)

(the powerset of the natural numbers with the Scott-topology)

(And there are several other characterizations...) Some basic facts: Every quasi-Polish space is sober. A metrizable space is quasi-Polish iff it is Polish. Every countably based locally compact sober space is quasi-Polish A subspace of a quasi-Polish space is quasi-Polish iff it is a Π0

2-subspace.

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Preliminaries (3)

Every quasi-Polish space X satisfies a generalized Baire category theorem (see R. Heckmann, 2012 and V. Becker & S. Grigorieff, 2012): Proposition If X =

i∈ω Ai, with each Ai a Σ0 2-set, then some Ai has

non-empty interior. Every quasi-Polish space X satisfies the Hausdorff-Kuratowski theorm: Proposition If A is a ∆0

2-subset of X, then there is a countable ordinal α and

an increasing sequence {Uβ}β<α of open subsets of X such that A =

• {Uβ \
• γ<β

Uγ | β < α, r(β) = r(α)}, where r(α) = 0 is α is even, and r(α) = 1 if α is odd.

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Quasi-Polish Noetherian spaces (1)

Theorem The following are equivalent for a sober Noetherian space X:

1 X has countably many points. 2 X is countably based. 3 X is quasi-Polish.

Proof: (1 ⇒ 2): By J. Goubault-Larrecq’s characterization of sober Noetherian spaces, X has the upper topology, so countability implies a countable basis. (2 ⇒ 1): Every open is compact, hence equal to a finite union of basic

• pens, so the topology is countable. Sobriety then implies countability.

(2 ⇒ 3) holds because sober Noetherian spaces are locally compact sober spaces, and (3 ⇒ 2) holds by definition.

Corollary A subspace of a countable sober Noetherian space is sober iff it is a Π0

2-subspace.

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Quasi-Polish Noetherian spaces (2)

Lemma Every covering of a quasi-Polish Noetherian space X by ∆0

2-sets

admits a finite subcovering. (Equivalently, the topology generated by the ∆0

2-sets is compact,

and clearly Hausdorff) Proof: Since X is countable we can assume the covering is

• countable. By the Baire category theorem, there is a ∆0

2-set A0 in

the covering such that its interior, U0, is non-empty. For n ≥ 0, if X = Un, then we repeat the same argument with respect to X \ Un to get a ∆0

2-set An+1 in the covering with

non-empty interior relative to X \ Un. Define Un+1 to be the union of Un and the relative interior of An+1. Then Un+1 is an

• pen subset of X which strictly contains Un.

Since X is Noetherian, eventually X = Un, and A0, . . . , An will yield a finite subcovering of X.

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Quasi-Polish Noetherian spaces (3)

Corollary Every infinite quasi-Polish Noetherian space X contains a singleton set which is not a ∆0

2-set.

(In other words, every quasi-Polish Noetherian TD-space is finite) Proof: Assume for a contradiction that every singleton is ∆0

2.

Then

x∈X{x} is a covering of X by ∆0 2-sets with no finite

• subcover. Contradiction.

(Note: Infinite non-sober Noetherian TD-spaces do exist)

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Quasi-Polish Noetherian spaces (4)

Lemma Every ∆0

2-subset A of a quasi-Polish Noetherian space X is

constructible.

Proof: By the Hausdorff-Kuratowski theorem, A =

• {Uβ \
• γ<β

Uγ | β < α, r(β) = r(α)} for some α < ω1 and increasing sequence {Uβ}β<α of open subsets of

• X. The Noetherian assumption implies we can take α to be finite.

(Note: This does not hold for non-sober Noetherian spaces)

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Closed sets

Theorem (J. Goubault-Larrecq, 2007) Every closed subset of a sober Noetherian space is finitely generated (equal to the closure of a finite set). Proof: (J. Goubault-Larrecq gave a general and elegant proof using de Groot duality. We just prove the quasi-Polish case) It suffices to show that each Noetherian quasi-Polish space X equals the closure of a finite set. Clearly, X =

x∈X cl({x}). Since closed sets are ∆0 2, from our

previous lemma there is finite F ⊆ X such that X =

x∈F cl({x}). Therefore, X = cl(F).

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Continuous images

Sobriety is not preserved under continuous images in general, but it is for Noetherian spaces: Lemma Continuous images of sober (quasi-Polish) Noetherian spaces are sober (quasi-Polish) Noetherian spaces.

Proof: Assume X is sober Noetherian and f : X → Y is a continuous

• surjection. Let C ⊆ Y be irreducible closed. Then f −1(C) is closed so

there is finite F ⊆ X such that cl(F) = f −1(C). Continuity implies cl(f(F)) ⊇ f(cl(F)) = C, hence cl(f(F)) = C. Since f(F) is finite and C is irreducible, C must equal the closure of some element of f(F). Therefore, Y is sober (and if X is countable then so is Y ). Theorem The category of sober (quasi-Polish) Noetherian spaces is finitely complete and finitely co-complete, but not cartesian closed.

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Main result

The following is a weak version of a theorem of Chevalley. Theorem (Quantifier elimination) Assume X and Y are quasi-Polish Noetherian spaces, and let P ⊆ X × Y be constructible. Then ∃XP := {y ∈ Y | (∃x ∈ X)P(x, y)}, and ∀XP := {y ∈ Y | (∀x ∈ X)P(x, y)} are constructible subsets of Y .

Proof: It suffices to prove the case for ∃XP because ∀XP = ¬∃X(¬P). As mentioned earlier, ∃XP is Σ0

2 because X is countable.

Furthermore, ∃XP is a quasi-Polish Noetherian space because it is the continuous image of the quasi-Polish Noetherian space P. It follows that ∃XP is a Π0

2-subset of Y .

Therefore, ∃XP is a ∆0

2-subset of Y , hence constructible.

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Conclusion

Assuming predicates are open and variables range over quasi-Polish Noetherian spaces, we showed that (classical) first order definable subsets are constructible The application of techniques from descriptive set theory in this context is interesting, but of course more constructive (and more informative) proofs are possible

When I showed the result to J. Goubault-Larrecq, he quickly replied with a nice combinatorial proof

To do: Better understand Chevalley’s theorem and its applications to quantifier elimination for the theory of algebraically closed fields.

It is unclear how applicative the result presented here is, but hopefully appropriate generalizations will lead to new perspectives on quantifier elimination

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