EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 1, 2024, 356-361 ISSN 1307-5543 – ejpam.com Published by New York Business Global Codimension one foliation and the prime spectrum of a ring Badr AlHarbi Department of Mathematics, Al Jumum University College, Umm Al-Qura University, , Saudi Arabia Abstract. Let F be a transversally oriented codimension-one foliation of class Cr, r ≥ 0, on a closed manifold M . A leaf class of a leaf F is the union of all leaves having the same closure as F . Let X be the leaf classes space and X0 be the union of all open subsets of X homeomorphic to R or S1. In [3, Theorem 3.15] it is shown that if a codimension one foliation has a finite height, then the singular part of the space of leaf classes is homeomorphic to the prime spectrum (or simply the spectrum) of unitary commutative ring. In this paper we prove that the singular part of the space of leaf classes is homeomorphic to the spectrum of unitary commutative ring if and only if every family of totaly ordered leaves is bounded below. 2020 Mathematics Subject Classifications: 54F65, 54H20 Key Words and Phrases: Foliation, prime spectrum, ring, the space of leaf classes 1. Introduction A space X is a spectral space [5] if it is (1) sober (i.e., every nonempty irreducible closed subset of Y is the closure of a unique point), (2) quasi-compact, (3) the quasi-compact open subsets of X form a basis and (4) the family of quasi-compact open subsets of X is closed under finite intersections. In particular a finite T0-space is a spectral space. (1), (2), (3) and (4) are called spectral properties. A foliation of codimension one on a smooth manifold M of dimension m is an open equivalence relation F on M with each equivalence class (called a leaf) is a weakly em- bedded m− 1 sub-manifold such as the canonical projection of M on the space of leaves M/F is a locally submersion. In that event for each x ∈ M , there is a chart (U,φ) such that φ(U) = Rm and each equivalence class of the restriction of F to U is homeomorphic to Rp × {y} where y ∈ R. Such chart (U,φ) is a distinguished chart. The notions of proper leaf, minimal set, local minimal set are introduced in [4, chapter 4.4]. DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4803 Email address: bhharbi@uqu.edu.sa (B. AlHarbi) https://www.ejpam.com 356 © 2024 EJPAM All rights reserved. B. AlHarbi / Eur. J. Pure Appl. Math, 17 (1) (2024), 356-361 357 Recently, we study the relationships between graphs and the prime spectrum of unitary commutative rings [1]. In [3] the authors studied relationships between foliations and spectral space. In par- ticular, if a codimension one foliation has a finite height, then the singular part of the space of leaf classes is homeomorphic to the spectrum of unitary commutative ring. In this paper we prove that the singular part of the space of leaf classes is homeomorphic to the spectrum of unitary commutative ring if and only if every family of totaly ordered leaves is bounded below. 2. Useful notions A topological space X is a T0-space (or Kolmogorov space) if for every x ̸= y, there is a neighborhood containing one of them but not the other; which is equivalent to the following implication ({x} = {y} ⇒ x = y). Let (X,≤) be an ordered set and T be a topology on X. We say that T is compatible with ≤ if, for each element x ∈ X, {x} = {y ∈ X : x ≤ y} = [x,→ [ ( {x} is the closure of {x}). Proposition 2.1. Let (X,≤) be an ordered set and T be a topology on X. If T is compatible with ≤, then (X, T ) is a T0-space. Proof. Let x and y be two points of X. • If x < y, then x ∈ X − [y,→ [ and so the open set X − [y,→ [ contains x and not contains y. • If x and y are not comparable, then the open set X − [x,→ [ contains y and not contains x. Remark 2.2. If (X, T ) is a T0-space, then X is an ordered set by the order defined by x ≤T y if and only if x ∈ {y}. According to [2] we have the following proposition: Proposition 2.3. If (X, T ) and (X ′, T ′) are two homeomorphic T0-spaces, then the or- dered sets (X,≤T ) and (X ′,≤T ′) are isomorphic. Proof. Let h be a homeomorphism between (X, T ) and (X ′, T ′). If x ≤T y, then x ∈ {y}T . Since h is continuous, h(x) ∈ {h(y)}T ′ and so h(x) ≤T ′ h(y). If now h(x) ≤T ′ h(y), then, by the continuity of h−1, x ≤T y. Therefore h is an isomorphism. The converse of the Proposition 2.3 is false. Indeed, all the compatible topologies with an order ≤ induce the same order ≤ but are not necessarily homeomorphic. A topological space X is quasi-compact if it satisfies the property of Borel-Lebesgue but it is not necessarily a Hausdorff space. A subset A of X is quasi-compact if it is a quasi-compact space equipped with the relative topology of X. We have the following properties: B. AlHarbi / Eur. J. Pure Appl. Math, 17 (1) (2024), 356-361 358 1. The quasi-compactness is invariant under continuous map. 2. Closed subsets of a quasi-compact space are quasi-compact. 3. The union of finitely many quasi-compact subsets is quasi-compact. The intersection of tow quasi-compact open subsets is not necessarily quasi-compact. The following example confirm this result: Example 2.4. In the two Euclidean space we consider the following points: C(0, 1), A(−1, 0), B(1, 0), An(−1, −1 n )) and Bn(1, −1 n )). Let X be the set {C,A,B,An, Bn : n ≥ 1} equipped with the following topology: {∅, X, U = {A,An : n ≥ 1}, V = {B,Bn : n ≥ 1}, Un = {Ap : n ≥ p ≥ 1}, Vn = {Bp : n ≥ p ≥ 1}}. Note that U and V are quasi-compact. Since U ∩ V = ⋃ n Un ∪ Vn and Un ∪ Vn is an increasing sequence of open subsets, U ∩ V is not a quasi-compact subset. A closed subset C is irreducible if it is not the union of two proper closed subsets or if the intersection of two nonempty open subsets is nonempty. An element x of C is called a generic point if the closure of the singleton {x} is equal to C: {x} = C. Let A be a commutative and unitary ring and Spec(A) be the set of prime ideals of A. If P is an ideal of A, the family {V (P ) = {Q ∈ Spec(A) : P ⊂ Q}} defines the closed subsets of the Zariski topology in Spec(A). Equipped with this topology, Spec(A) satisfies the following properties: 1. Spec(A) is Kolmogorov. 2. Every irreducible closed subset of Spec(A) has a generic point. 3. Spec(A) is a quasi-compact space. 4. There is a basis of quasi-compact open subsets of Spec(A). 5. The family of quasi-compact open subsets of Spec(A) is stable under finite intersec- tion. If a topological space X satisfies the above five properties, then there exists a com- mutative and unitary ring A such that X is homeomorphic to Spec(A) equipped with the Zariski topology [5]. Let F be a codimension-one transversally oriented foliation of class Cr, r ≥ 0, on a closed m-manifold M . Dippolito [4, chapter 4.4] defined the boundary δU of a nonempty saturated connected open subset distinct of M . The boundary δU is equal to the set of points x ∈ M − U such as there is a curve c : [0, 1] → M such that c(0) = x and c(]0, 1]) ⊂ U . Dippolito proved that δU is a union of a finitely many leaves ([4, chapter 4.4]). We have δU = U − U . Note that an attracting proper leaf from one side is introduced in [4, chapter 4.4]. Recall that if A ⊂ X, the saturation SatR(A) of A is the union of all equivalence classes meeting A. The subset A is called invariant (or saturated) if A = SatR(A). Note B. AlHarbi / Eur. J. Pure Appl. Math, 17 (1) (2024), 356-361 359 that, the interior, the closure, the boundary of each saturated subset is also saturated; indeed the relation R is open. We denote by Ts the invariant topology on X formed by the invariant open subsets of X. An invariant open subset U ⊂ X is called compact by saturation if it is quasi-compact for the invariant topology Ts. That is, every covering (Ui) of U by invariant open subsets Ui contains a finite sub-cover. Lemma 2.5. Let R be an open equivalence relation on a topological space X. An open subset V of the quotient space X/R is quasi-compact if and only if the open subset U = q−1(V ) is compact by saturation. Proof. Suppose that V is quasi-compact, and let (Ui, i ∈ I) be a covering of U = q−1(V ) by saturated open subsets. Thus the open subsets (q(Ui)) cover V , and some finite number of these, q(Ui1), ..., q(Uin), covers V . Because every Ui is saturated, q−1(q(Ui)) = Ui and hence U = Ui1 ∪ ... ∪ Uin . Conversely, let (Vi, i ∈ I) be a family of open subsets of X/R such that V = ⋃ i Vi. Since q−1(Vi) is a saturated open subset and U is compact by saturation, it follows that U = q−1(Vi1) ∪ ... ∪ q−1(Vin) which implies that V = Vi1 ∪ ... ∪ Vin . Lemma 2.6. [3] Let U be a connected nonempty invariant open subset of M . Then the following properties are equivalent: a) The following two properties hold: i) Each leaf L ⊂ δϵU , ϵ = ±, is attracting from the side ϵ (i.e L is attracting from the side of U). ii) For each leaf F ⊂ U , the intersection F ∩ U contains a local minimal set in U . b) U is compact by saturation. Lemma 2.7. [4, chapter 4.4] For each leaf F of a nonempty invariant open subset U ⊂ M , the intersection F ∩ U contains at most finitely many local minimal sets in U . Let X = M/F̃ be the leaf classes space. Consider X0 the union of all open subsets of X homeomorphic to R or S1. Proposition 2.8. [3] The inverse image of X0 by the canonical projection p is the union of all stable proper leaves. Bouacida et all showed, in [3], that if F has a well defined height, then the singular part X−X0 of the leaf classes space is homeomorphic to the spectrum of a unitary commutative ring equipped with the Zariski topology. Note that, the height of a foliation is well defined [4, chapter 4.4] if and only if every totally ordered family of leaves is well-ordered (i.e. it has a minimal element). Recall that a family of leaves is ordered by inclusion of their closures. Precisely, the authors of [3] showed that X −X0 verifies the following properties: (1) X −X0 is a sober space. B. AlHarbi / Eur. J. Pure Appl. Math, 17 (1) (2024), 356-361 360 (2) X −X0 is a quasi-compact space. (3) X −X0 has a basis of quasi-compact open subsets. (4) If F has a well defined height, then the family of quasi-compact open subsets of X −X0 is closed under finite intersections. In this paper we prove that the singular part of the space of leaf classes is homeomorphic to the spectrum of unitary commutative ring if and only if every family of totaly ordered leaves is bounded below (Theorem 3.1). 3. Main result Theorem 3.1. Let F be a codimension-one transversally oriented foliation of class Cr, r ≥ 0, on a closed manifold M . Consider X the leaf classes space and let X0 be the union of all open subsets of X homeomorphic to R or S1. Then, the space X − X0 is homeomorphic to the spectrum of unitary commutative ring if and only if every family of totaly ordered leaves is bounded below. We need the following lemmas. Lemma 3.2. [3] Let F be a codimension-one transversally oriented foliation of class Cr, r ≥ 0, on a closed manifold M . Consider X the leaf classes space and let X0 be the union of all open subsets of X homeomorphic to R or S1. Then, e get the following properties: (1) The space X −X0 is sober. (2) The space X −X0 is quasi-compact. (3) The space X −X0 has a basis of quasi-compact open subsets. Proof. of Theorem 3.1. If X −X0 is homeomorphic to the prime spectrum of unitary commutative ring, then it satisfies the condition (K1) of Kaplansky and so every totally ordered family of orbits has an infimum. By Lemma 3.2, X −X0 satisfies three spectral properties (1), (2) and (3). It suffices to show the fourth spectral property, that is, if every family of totaly ordered leaves is bounded below, then the family of quasi-compact open subsets of X −X0 is closed under finite intersections. According to Lemma 2.6, it suffices to show that the intersection W = U ∩ V of two compact by saturation open sets is also compact by saturation. According to that fact that δϵW ⊂ δϵU ∪ δϵV , ϵ = ±, we prove that W verifies the property a − ii) of Lemma 2.6. We can suppose that W is a connected set, differently we can take the connected component of W containing a leaf F ⊂ W . Consider {Fi} a maximal totally ordered family of leaves such that Fi ⊂ F ∩W , for every i, and we denote by L the greatest lower bound leaf of this family (this leaf L exists from the hypothesis). According to Lemmas 2.7 and 2.6-a-ii), there exist two local minimal REFERENCES 361 sets E1 and E2 of F restricted to U and V respectively which are subsets of the closure Fi, for every i. Consider L1 and L2 two leaves such as L1 ⊂ E1 and L2 ⊂ E2. Therefore L1 = E1 ⊂ L and L2 = E2 ⊂ L. Consequently, L ⊂ U and L ⊂ V , thus L ⊂ W and L∩W is a local minimal set of F restricted to W . Differently, there is a leaf S such that S ̸= L and S ⊂ L ∩ W . Thus the family {Fi} is not maximal which leads to a contradiction. We deduce that the open set W verifies the two items of the property a) in Lemma 2.6. Therefore W it is a compact by saturation open set. This ends the proof of Theorem 3.1. References [1] B. Alharbi. Graphs and the prime spectrum of unitary commutative rings. European Journal of Pure and Applied Mathematics, 16:314–318, 2023. [2] E. Salhi E. Bouacida, O. Echi. Foliations, spectral topology and special morphism. Comm.Ring.Theory III Lect.not.Pure.Appl.Math.M.Dekker, 205:111–132, 1999. [3] E. Salhi E. Bouacida, O. Echi. Foliation and spectral topology. J. Math. Soc. Japan, 52:447–464, 2000. [4] C. Godbillon. Feuilletages. Etudes géométriques. Birkauser -Verlag, New York, 1991. [5] M. Hochster. Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142:43–60, 1969.