Alexandroff Topology of Algebras over an Integral Domain
Shai Sarussi

TL;DR
This paper introduces a topology on the set of $S$-nice subalgebras of an $F$-algebra over an integral domain, enabling the study of their algebraic structure through topological methods.
Contribution
It defines an Alexandroff topology on $S$-nice subalgebras and characterizes irreducible subsets and components within this topological space.
Findings
Irreducible subsets have a supremum in the space.
Characterization of irreducible open sets.
Description of irreducible components.
Abstract
Let be an integral domain with field of fractions and let be an -algebra. An -subalgebra of is called -nice if is lying over and the localization of with respect to is . Let be the set of all -nice subalgebras of . We define a notion of open sets on which makes this set a Alexandroff space. This enables us to study the algebraic structure of from the point of view of topology. We prove that an irreducible subset of has a supremum with respect to the specialization order. We present equivalent conditions for an open set of to be irreducible, and characterize the irreducible components of
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Alexandroff Topology of Algebras over an Integral Domain
Shai Sarussi
Abstract
Let be an integral domain with field of fractions and let be an -algebra. An -subalgebra of is called -nice if is lying over and the localization of with respect to is . Let be the set of all -nice subalgebras of . We define a notion of open sets on which makes this set a Alexandroff space. This enables us to study the algebraic structure of from the point of view of topology. We prove that an irreducible subset of has a supremum with respect to the specialization order. We present equivalent conditions for an open set of to be irreducible, and characterize the irreducible components of .
1 Introduction and some preliminary results
As the title suggests, in this paper we discuss algebras over integral domains from the point of view of Alexandroff topology, which will shortly be defined. In [Sa1] and [Sa2] we studied algebras over valuation domains, concentrating on quasi-valuations that extend a valuation on a field. In [Sa3] we prove several existence theorems of integral domains that may be applied to the study of quasi-valuations. More specifically, let be an integral domain which is not a field, let be its field of fractions, and let be an -algebra. In [Sa3] we study -subalgebras of , that are lying over and whose localizations with respect to is . We call them -nice subalgebras of . Namely, an -subalgebra of is called an -nice subalgebra of if and ; we shall use this notation throughout this paper. We denote by the set of all -nice subalgebras of .
We recall now some definitions and results from [Sa3]. The following concept is used quite frequently: let be a basis of over . We say that is -stable if there exists a basis of over such that for all and , one has .
We note in [Sa3, Remark 3.4] that if a basis is closed under multiplication then is -stable. Thus, for example, every free (noncommutative) -algebra with an arbitrary set of generators has an -stable basis; in particular, every polynomial algebra with an arbitrary set of indeterminates over has an -stable basis.
We also show in [Sa3, Proposition 3.12] that if is finite dimensional over , then every basis of over is -stable. The first existence theorem is as follows.
Theorem 1.1**.**
(cf. [Sa3, Theorem 3.14]) If there exists an -stable basis of over , then there exists an -nice subalgebra of .
In particular, if is finite dimensional over then there exists an -nice subalgebra of .
The following result is a going-down lemma for -nice subalgebras.
Lemma 1.2**.**
(cf. [Sa3, Lemma 3.20]) Let be integral domains with field of fractions such that . Assume that there exists an -stable basis of over . Let be an -nice subalgebra of . Then there exists an -nice subalgebra of , which is contained in .
We conclude that a minimal -nice subalgebra of does not exist. More precisely, we prove
Proposition 1.3**.**
(cf. [Sa3, Proposition 3.21]) Assume that there exists an -stable basis of over . Let be an -nice subalgebra of . Then there exists an infinite decreasing chain of -nice subalgebras of starting from . In particular, a minimal -nice subalgebra of does not exist.
Definition 1.4**.**
Let be a chain of prime ideals of and let be a faithful -algebra. Let be a chain of prime ideals of . We say that covers if for every there exists lying over ; namely, .
Theorem 1.5**.**
(cf. [Sa3, Theorem 3.24]) Assume that there exists an -stable basis of over . Let be a chain of prime ideals of . If is commutative then there exists an -nice subalgebra of such that there exists a chain of prime ideals of covering . In fact, there exists an infinite descending chain of such -nice subalgebras of .
Note that by Theorem 1.1 and Proposition 1.3 if contains an -stable basis then is not empty and is, in fact, infinite. In particular, by the above-mentioned note before Theorem 1.1, if is finite dimensional over then is infinite.
In the second section of this paper, which is its main part, we do not assume that contains an -stable basis; we merely assume that is not empty. We do, however, present an example in which the existence of an -stable basis is assumed. In the third section of this paper we assume that contains an -stable basis
We present now some of the common definitions we use from order theory. Let be a set. A relation on that is reflexive and transitive is called a quasi order or a preorder; if the relation is also antisymmetric then it is called a partial order and is called a partially ordered set, or a poset. Let . We say that is a lower (resp. upper) bound of if (resp. ) for all . If the set of lower (resp. upper) bounds of has a unique greatest (resp. smallest) element, this element is called the greatest lower (resp. least upper) bound of , and is denoted by (resp. ). We say that is a lower set if . We say that is an upper set if . A subset is called directed if for all there exists such that and . We say that is a dcpo (directed complete partial order) if every directed subset of has a supremum. A subset of is called an ideal of if is a lower set and directed. is called an inf semilattice (resp. sup semilattice) if for all , (resp. ) exists in . If is both an inf semilattice and a sup semilattice, we say that is a lattice. A subset of is called a sublattice of if is a lattice with respect to the partial order of . Similarly, one defines inf subsemilattice and sup subsemilattice.
We briefly discuss now the notion of an Alexandroff topological space. A topological space whose set of open sets is closed under arbitrary intersections is called an Alexandroff space, after P. alexandroff who first introduced such topological spaces in his paper [Al] from 1937. Equivalently, A topological space is called an Alexandroff space if every element has a minimal open set containing it. A finite topological space is the most important particular case of an Alexandroff space. In fact, Alexandroff spaces share many properties with finite topological spaces; in particular, Alexandroff spaces have all the properties of finite spaces relevant for the theory of digital topology (see [He] and [Kr]). Thus, in the eighties the interest in Alexandroff spaces arose as a consequence of the very important role of finite spaces in digital topology. In 1999 F. G. Arenas studied the topological properties of Alexandroff spaces (see [Ar]).
Let be a topological space. For we denote by the closure of . It is well known and easy to prove, that if one defines whenever , then is a quasi order; i.e., a reflexive and transitive relation. is called the specialization order. Recall that is called if for every two distinct elements in , there exists an open set containing one of them but not the other. It is known that if is then is a partial order. On the other hand, for any quasi order on a set , one can define the topology whose open sets are the upper sets of with respect to , denote it by . So, there are two functors, the specialization order from the class of all topological spaces to the class of all quasi ordered sets, sending to ; and the functor in the opposite direction sending to . If one restrict the class of all topological spaces to the the class of all alexandroff topological spaces, then one has an isomorphism between the categories.
In this paper the symbol means proper inclusion and the symbol means inclusion or equality.
2 The Alexandroff Topology
Inspired by the Zarisky topology on the prime spectrum of a ring, for every we denote by the set of all -nice subalgebras of containing . It is easy to see that , , and for every , we have
[TABLE]
Thus, the set B= is a basis for a topology on . Namely, every open set in is a union of elements of . Moreover, for every set of subsets of , we have
[TABLE]
Now, an intersection of union of elements of can be presented as a union of intersection of elements of . Indeed, let be a set and let be a set of sets such that for all and there exists a set ; then
[TABLE]
where denotes the -th component of . We apply the above equation to elements of and deduce that every intersection of open sets of is open. Thus, is an Alexandroff topological space with respect to the topology defined above.
Let be a topological space and let ; then iff for every open set containing , we have . If is alexandroff, this is equivalent to , where denotes the minimal open set containing . Now, let . It is easy to see that the minimal open set containing is . Indeed, let be an open set containing ; then is of the form where for all . Thus, for some ; hence and . On the other hand, is an open set containing . Thus, the specialization order on is the order of inclusion; i.e., for , iff . Moreover, as in any alexandroff topological space, is open iff is an upper set with respect to the specialization order; dually, is closed iff is a lower set.
We will frequently use the following four basic lemmas.
Lemma 2.1**.**
1. Let and be two elements of and let be an -algebra satisfying ; then is an -nice subalgebra of . 2. Let be a finite subset of ; then .
Proof.
Straightforward.
∎
For subsets and we define
[TABLE]
Lemma 2.2**.**
Let be a nonempty subset of and denote . Then the following three properties are valid:
(a) ;
(b) ; and
(c) If is closed under addition then .
In particular, if is a ring then it is an -nice subalgebra of .
Proof.
(a) Clearly, . Let ; then for some . Since , we have . (b) Since is not empty there exists such that ; thus, . (c) Let and let ; then for all , for appropriate . Thus, for all , ; hence, . ∎
Note that is not necessarily an -algebra since it is not necessarily a ring.
Lemma 2.3**.**
Let be a nonempty chain in . Then the supremum of exists in .
Proof.
Let ; since is a chain, is ring. By Lemma 2.2, . It is clear that is the smallest -nice subalgebra of containing every element of .
∎
Lemma 2.4**.**
Let be a nonempty open set of . Then there exists a maximal element in , which is also a maximal element in .
Proof.
Consider with the partial order of containment. Let be a nonempty chain in . By Lemma 2.3, exists in . Now, for some ; thus, by the definition of an open set in , . Therefore, by Zorn’s Lemma, there exists a maximal element in .
∎
Remark 2.5*.*
In view of Lemma 2.4, one can be more precise. In fact, for every maximal chain in a nonempty open set of (there exists such a chain by Zorn’s Lemma), is a maximal element of , and . Moreover, for the same reason, for any there exists a maximal element containing ; indeed, take any maximal chain in containing , and use the reasoning above.
Let be an open set. We consider the following properties:
(a) is of the form for some .
(b) is closed under arbitrary nonempty intersection.
(c) .
(d) .
(e) is closed under finite nonempty intersection; i.e., is an inf subsemilattice of .
(f) contains no more than one minimal element.
Proposition 2.6**.**
Notation as above; the following implications hold:
.
Proof.
. By definition, every element of contains . Let where . Then for some ; hence by Lemma 2.1, and thus, by the definition of , . and . By assumption , which is clearly the smallest element of ; since every contains and is an open set, we get . is trivial. . As above, every contains ; thus, . . Let ; by Lemma 2.1, ; clearly, , and thus . . Assume to the contrary that there exists two different minimal elements in . Then, by assumption , but it is clearly strictly contained in both and , a contradiction.
∎
We present now examples which demonstrate that the left to right implications in the previous proposition cannot be reversed.
Example 2.7*.*
To show that the implication may not hold, we can consider any case in which is taken as the open set and ; it is clear that . As an explicit example, let , and . More generally, in [Sa3, discussion after Proposition 3.21] we showed that whenever contains an -stable basis, we have .
To show that the implication may not hold, one can consider an infinite chain of -nice subalgebras of such that their intersection is an -nice subalgebra of that is strictly contained in each of them. As an explicit example, let be a valuation domain with value group , and let , where is the field of fractions of . Take and for all let . For all denote
[TABLE]
where is any nonzero ideal of . Let . Since is a chain, is closed under finite nonempty intersection; however,
[TABLE]
Finally we demonstrate that does not necessarily imply . With the notation presented above, let be an ideal of strictly containing ; and let
[TABLE]
Let . Then the unique minimal element of is , but in not closed under finite nonempty intersection since for all , we have ; thus,
Proposition 2.8**.**
Let be a nonempty subset of ; then
1. There exists a lower bound for iff the infimum of exists.
2. There exists an upper bound for iff the supremum of exists.
Proof.
Clearly, the right to left implication in both statements is trivial. We prove the left to right implication of the first statement. Let be a lower bound of , let and let be any element of (note that is not empty). Then by Lemma 2.1, and it is clearly the infimum of . We prove now the left to right implication of the second statement. Let denote the set of all upper bounds of ; by assumption, is not empty. Since is not empty, has a lower bound; indeed, any element of is a lower bound for . Thus, by the first statement, the infimum of , , exists in . Now, let , then for every , we have ; thus, . Therefore, and is the supremum of .
∎
We note that the assumption that is crucial. Indeed, the empty set clearly has an upper bound but the supremum of does not exist, since may not contain a smallest element, as shown in Example 2.7. Also, the empty set clearly has a lower bound but the infimum of exists iff is irreducible, as we shall see in Theorem 2.11.
Dually to Proposition 2.6, let be a closed set. We consider the following properties:
(a) for some .
(b) Every nonempty subset of has a supremum, which belongs to .
(c) and .
(d) and .
(e) is a sup subsemilattice of .
(f) contains no more than one maximal element.
The proof of the following proposition is quite similar to the proof of Proposition 2.6. The implications , , and rely on Proposition 2.8; we shall not prove it here.
Proposition 2.9**.**
Notation as above; the following implications hold:
.
Note the small difference in the implication , which is valid in Proposition 2.9 but not in Proposition 2.6. The reason for this difference is the fact that is defined for all whereas is defined only for . While in condition (c) of Proposition 2.6, is not necessarily in , in conditions (c) and (d) of Proposition 2.9 we require that would be in .
We also note that condition (e) of Proposition 2.9 implies that is an ideal of , in the sense of order theory defined in the introduction. Indeed, by assumption is closed and thus it is a lower set. By the assumption in (e), is a sup subsemilattice of ; in particular, is directed. In fact, in this case is actually a sublattice of . To show this, let , the infimum of and is their intersection which is in ; by assumption is a closed set and thus it is a lower set, so ; and by the assumption in condition (e), is a sup subsemilattice of .
The following theorem is important to our study.
Theorem 2.10**.**
Let be an irreducible subset of . Then is an -nice subalgebra of ; in particular, .
Proof.
Denote . Let ; we prove that there exists such that . Assume to the contrary that there exists no such . Let denote the set of all elements in not containing , and let denote the set of all elements in not containing . It is clear that and are closed in . However, by our assumption , while and , a contradiction. Thus, is a ring. By Lemma 2.2, .
∎
Theorem 2.11**.**
Let be a nonempty open set. The following statements are equivalent:
(a) has a greatest element.
(b) There exists such that .
(c) has a unique maximal element.
(d) is irreducible.
(e) is a sup subsemilattice of .
(f) Every nonempty subset has a supremum.
Proof.
We prove . To show that we denote by the greatest element of . It is clear that every closed set containing also contains ; on the other hand, since is the greatest element of , by the definition of the topology on , every closed set containing also contains . . We prove that is the unique maximal element of . It is clear that is a maximal element of , since otherwise and thus . Similarly, Assuming there exists another maximal element , we get . . Let denote the unique maximal element of . Assume to the contrary that where are closed in while and . Let and . By Remark 2.5, there exist maximal elements containing and , respectively. Since is the unique maximal element of , we have . Thus, and , a contradiction. We prove . Let ; by assumption is irreducible and thus by Theorem 2.10, is an -nice subalgebra of . Clearly, contains both and . Thus, by Proposition 2.8, exists. Hence, by the definition of an open set, . We prove now . Let . By Remark 2.5, for every there exists a maximal element containing . By assumption, every two elements of have a supremum, thus these must all be equal. So, is bounded from above and therefore, by Proposition 2.8, the supremum of exists. Finally, we show . By assumption has a supremum, and since is an open set, its supremum is its greatest element.
∎
In view of Theorem 2.11, we characterize now the irreducible components of .
Proposition 2.12**.**
* is an irreducible component of iff for some maximal .*
Proof.
Let be a maximal element of . It is clear that is irreducible. Assume to the contrary that there exists an irreducible set . Let . Then and . By Theorem 2.10, . However, strictly contains , a contradiction. On the other hand, let be an irreducible component of . Let ; by Theorem 2.10, . Thus, . Since is irreducible and is an irreducible component of , one has . Now, is maximal in , since otherwise there exists , but then , a contradiction.
∎
Recall from [GHKLMS, Def. O-5.6.] that a topological space is called sober if for every irreducible closed subset of , there exists a unique such that is the closure of ; i.e., has a unique generic point. Also recall that a poset is called dcpo (directed complete partial order) if every directed subset of has a supremum. It is known (see, for example [GHKLMS, Ex. O-5.15.]) that every sober space is a dcpo, under the specialization order.
Note that by Lemma 2.4 there exists a maximal -nice subalgebra of . In [Sa3, discussion after Corollary 3.21] we noted that in case is a valuation domain of and is a field, then the maximal -nice subalgebras of are precisely the valuation domains (whose valuations extend ) of . So, by Proposition 2.12, the closures of these valuation domains are precisely the irreducible components of .
We also showed in [Sa3, Example 3.26] that even in the case of a central simple -algebra, one can have an infinite ascending chain of -nice subalgebras of (even when is a valuation domain). For the reader’s convenience we present here the example.
Example 2.13*.*
(cf. [Sa3, Example 3.26]) Let be a non-Noetherian integral domain with field of fractions . Let be an infinite ascending chain of ideals of and let . Then
[TABLE]
is an infinite accending chain of -nice subalgebras of . So, let
[TABLE]
then is closed and irreducible with no generic point.
In particular, we have an example in which is not sober. Nevertheless, in a subsequent paper we will show that is indeed a dcpo and has some interesting properties from the point of view of domain theory.
Remark 2.14*.*
Note that by Lemma 2.1, is an inf semilattice, where the infimum is actually an intersection of sets. In particular, taking in Theorem 2.11, the conditions of the theorem are also equivalent to the condition that is a lattice. In fact we can say even more. By the definition of an irreducible space, every finite intersection of nonempty open sets is nonempty. In our case, whenever satisfies the equivalent conditions of Theorem 2.11, the intersection of all nonempty open sets of is the singleton , where denotes the greatest element of . Moreover, this property is also equivalent to the equivalent conditions presented in Theorem 2.11.
Recall (cf. [En, Theorem 16.4]) that a valuation on a field is called henselian if extends uniquely to every algebraic field extension of ; in this case, one also says that the corresponding valuation domain is henselian. Thus, in view of the previous remark and the discussion before Example 2.13, we have,
Example 2.15*.*
If is an henselian valuation domain, is its field of fractions and is an algebraic field extension of , then is a lattice (viewed from the point of view of order theory), and an irreducible topological space (viewed as an Alexandroff topological space).
3 Prime ideals
In this short section we discuss the prime spectra of -nice subalgebras of and the subsets of covered by them. We assume that is commutative (in this case whenever are -subalgebras of and is a prime ideal of , then is a prime ideal of ). We also assume that contains an -stable basis, in order to be able to use the going down lemma for -nice subalgebras of (Lemma 1.2) and Theorem 1.5.
For a ring , we denote by the prime spectrum of ; i.e., the set of all prime ideals of . Recall the following definition from [Sa3]: if for every there exists lying over , we say that satisfies “Lying Over” (LO, in short) over . We denote by the set of all prime ideals of having a prime ideal of lying over them; namely,
there exists lying over . Note that, by definition iff does not satisfy LO over .
As usual, we use the term “almost all” to mean that a property is satisfied to all but finitely many elements.
In the following lemma we show that whenever does not satisfy LO over , there exists an -nice subalgebra of whose prime spectrum lies over a larger set of primes of than the prime spectrum of .
Lemma 3.1**.**
Let be an -nice subalgebra of such that , and let . Then there exists such that and .
Proof.
Consider the chain . By Theorem 1.5, there exists having a prime ideal lying over . By Lemma 2.1, is an -nice subalgebra of . Now, since is commutative, every prime ideal of () intersect to a prime ideal of . Thus, with .
∎
Proposition 3.2**.**
There exists satisfying LO over iff there exists whose prime spectrum is lying over almost all prime ideals of ; namely, is finite.
Proof.
is trivial; just take . Let be an -nice subalgebra of whose prime spectrum is lying over almost all prime ideals of . If satisfies LO over then we are done. Otherwise, by Lemma 3.1 there exists with . After finitely many such steps we get that satisfies LO over .
∎
Remark 3.3*.*
Note that if there exists that satisfies LO over , then for every nonempty closed set there exists that satisfies LO over ; indeed, take any and denote .
In view of the previous remark, in the following proposition we present equivalent conditions for the non-existence of an -nice subalgebra of that satisfies LO over .
Proposition 3.4**.**
The following conditions are equivalent:
(a) There exists no that satisfies LO over .
(b) There is no maximal subset (with respect to inclusion) such that there exists with .
(c) For every there exists an infinite descending chain of -nice subalgebras of such that for all , and , whenever .
(d) Every nonempty closed subset of contains an infinite descending chain of -nice subalgebras of such that , whenever .
Proof.
We prove . To show , assume to the contrary that there exists such that is a maximal subset of . By assumption . By Lemma 3.1 there exists such that , a contradiction to the maximality of . We prove now . Let . By assumption , since otherwise is a maximal subset of . By Lemma 3.1 there exists such that . By assumption ; so again by Lemma 3.1, there exists such that . We continue this way to get an infinite descending chain of -nice subalgebras of such that , whenever . To prove that , let be a nonempty closed subset of and let . The result follows by applying the assumption on and recalling the definition of the topology on . Finally, we show . Assume to the contrary that there exists satisfying LO over . Then the closed subset contains only elements that satisfy LO over , a contradiction.
∎
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