Asymptotic for the number of star operations on one-dimensional Noetherian domains
aa r X i v : . [ m a t h . A C ] S e p ASYMPTOTIC FOR THE NUMBER OF STAROPERATIONS ON ONE-DIMENSIONAL NOETHERIANDOMAINS
DARIO SPIRITO
Abstract.
We study the set of star operations on local Noether-ian domains D of dimension 1 such that the conductor ( D : T )(where T is the integral closure of D ) is equal to the maximal idealof D . We reduce this problem to the study of a class of closureoperations (more precisely, multiplicative operations) in a finiteextension k ⊆ B , where k is a field, and then we study how thecardinality of this set of closures vary as the size of k varies whilethe structure of B remains fixed. Introduction
Star operations are a class of closure operations on the set of (frac-tional) ideals of an integral domain that has been first studied by Krull[12] and Gilmer [4, Chapter 5]. Along the years, they have been used,for example, to study and generalize factorization properties (for exam-ple, with the characterization of unique factorization domains amongKrull domain through the t -class group by Samuel [16]) and to findPr¨ufer-like classes of integral domains (for example P v MDs; see forexample [1, Section 3] for an overview).More recently, Houston, Mimouni and Park have been studying theset Star( D ) of star operations on an integral domain D , attemptingto characterize when this set is finite and, in this case, to calculate itscardinality [7, 8, 9, 10]. They analyzed in particular the Noetheriancase, showing that | Star( D ) | = ∞ when D has dimension greater than1 [8, Theorem 2.1], showing how to reduce to the local case [8, Theorem2.3] and calculating the cardinality of Star( D ) when ℓ D ( T / m D ) ≤ T is the integral closure of D and m D the maximal ideal of D [8,Theorem 3.1]. Further cases have been considered, for example, in [8](for infinite residue field), in [15, 18] (for pseudo-valuation domains), in[17] (for Kunz domains) and [22] (for some numerical semigroup rings).Beyond star operations, that are two other classes of closure oper-ations that are in use in commutative algebra: semiprime operations,defined on the set of integral ideals of an arbitrary ring, and semistar Date : October 5, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Star operations; multiplicative operations; Noetheriandomains. operations, defined on the set of all submodules of the quotient field ofan integral domain. There are several connections between the classesof semiprime, star and semistar operations; indeed, most definitionsand many properties can be stated in essentially the same way, withthe main difference being often in how they must be phrased to accountfor the different partially ordered set on which the closures are defined.To unify the treatment of these classes of closure operations, the pa-per [19] introduced the concept of multiplicative operations : these area class of closure operations that can be defined in any ring extension A ⊆ B over any set G of A -submodules of B , and their definition isflexible enough to cover (for suitable choices of A , B and G ) all threeclassical cases. Furthermore, multiplicative operations enjoy some func-torial properties: while these are usually a staple of many semiprimeoperations, for star and semistar operations they are very rare, espe-cially due to the fact that quotienting an integral domain disrupts itsquotient field. (There are some limited exceptions: see [3] for proper-ties of star operations along pullbacks and [20] for an application toPr¨ufer domains.) In the special case of local Noetherian domains ofdimension 1, multiplicative operations allow to bypass this problem byconsidering only the submodules contained in the integral closure T ofthe starting domain D (which contain all the needed data), and thenby quotienting T over the conductor ( D : T ) (see [19, Section 6] andthe discussion after Definition 3.4 below). In particular, this reducesthe problem of finding all star operations on D to the study of theArtinian ring T / ( D : T ).In this paper, we continue the study initiated in Sections 6 and 7of [19] on this case, in particular concentrating on the case where theconductor ( D : T ) is equal to the maximal ideal m D of D ; in the Ar-tinian setting, this case correspond to the study of a particular set ofmultiplicative operations defined on a finite extension k ⊆ B , where k is a field. As we are interested in cardinality problems, we assumethroughout the paper that k is a finite field of cardinality q : in par-ticular, we are interested in what happens when the structure of B is“fixed” (see Section 3 for a more precise definition) while q changes,that is, we are interested in the cardinality of the set Star( D ) of staroperations on D as a function of q , and especially in understandinghow fast the growth of | Star( D ) | is.The aim of this paper is to introduce and give some evidence to thefollowing conjecture: if the structure of B is fixed, then the function q log q log | Star( D ) | has always a limit as q → ∞ , and this limitonly depend on the length of B = T / m D as a D -module. While weare not able to prove this conjecture in full generality, we show poly-nomial bounds on the upper and lower limits (respectively, Theorem4.2 and Proposition 5.8, summarized in Theorem 5.9) and we prove SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 3 the conjecture in full for ℓ D ( T / m D ) ≤ Closure operations
Let ( P , ≤ ) be a partially ordered set. A closure operation on P is amap c : P −→ P such that, for every x, y ∈ P : • x ≤ c ( x ); • if x ≤ y , then c ( x ) ≤ c ( y ); • c ( c ( x )) = c ( x ).If c, d are closure operations on P , we write c ≤ d if c ( x ) ≤ d ( x ) forevery x ∈ P .Any closure operation c is uniquely determined by the set P c := { x ∈ P | c ( x ) = c } of its fixed points, i.e., two closure operations c and d are equal if and only if P c = P d . Furthermore, c ≤ d if and only if P c ⊇ P d .Let A ⊆ B be a ring extension, and let G be a set of A -submodulesof B . A multiplicative operation [19] on ( A, B, G ) is a closure operation ⋆ on G , I I ⋆ , such that ( I : b ) ⋆ ⊆ ( I ⋆ : b )for all I ∈ G and all b ∈ B such that ( I : b ) ∈ G . We denote the set ofmultiplicative operations on ( A, B, G ) by Mult( A, B, G ).If D is an integral domain with quotient field K , a star operation on D is a closure operation ⋆ on the set F ( D ) of fractional ideals of D such that, for all I ∈ F ( D ) and all x ∈ K , we have: • D = D ⋆ ; • ( xI ) ⋆ = x · I ⋆ .We denote by Star( D ) the set of star operations on D . The restric-tion from F ( D ) to the set I ( D ) • of all nonzero ideals of D gives anisomorphism of ordered sets between Star( D ) and Mult( D, K, I ( D ) • )[19, Proposition 3.4]. 3. Artinian rings
Throughout the paper, we shall denote by k the finite field of cardi-nality q and by B a finite k -algebra that is a principal ideal ring; wealso write F q e for the field of cardinality q e , i.e., for the extension of k of degree e . In particular, F q = k .We can write B as a direct product B × · · · × B t , where each B i is alocal k -algebra; by [11, Theorem 8] B i is isomorphic to F q ei [ X ] / ( X f i ) ≃ F q ei [[ X ]] / ( X f i ) for some positive integers e i , f i . We will always consider k as a subring of B through the diagonal embedding; we shall alsosometimes consider k as a subring of B i by the obvious embedding of k into F q ei [ X ] / ( X f i ). DARIO SPIRITO
It is easy to see that the quantities e i , f i are linked to the length of B . Proposition 3.1.
Preserve the notation above. Then:(a) f i = ℓ B i ( B i ) = ℓ B ( B i ) ;(b) L i is the residue field of B i ;(c) ℓ A ( B ) = P i e i f i = ℓ D ( T / ( D : T )) .Proof. Straightforward. (cid:3)
Let s := { ( e , f ) , . . . , ( e t , f t ) } be a sequence of pairs of positive in-tegers. We call the ring B s := F q e [ X ]( X f ) × · · · × F q ek [ X ]( X f k )the Artinian ring associated to s . We denote by n ( s ) the sum P i e i f i = ℓ k ( B s ). Remark 3.2.
The methods we will use actually hold in the more gen-eral setting of a ring extension A ⊆ B , where A and B are Artinianrings and B is a finite A -module; the restriction to A = k , however,allows to simplify the notation. Furthermore, many proofs do not reallyuse the hypothesis that k is finite if not to guarantee that the quantitiesinvolved are finite.Let now k, B as above. We define: • F ( k, B ) := { I ∈ F k ( B ) | IB = B } ; • F reg ( k, B ) := { I ∈ F k ( B ) | I contains a regular element of B } ; • F ( k, B ) := { I ∈ F k ( B ) | ∈ I } .These sets are related in the following way. Proposition 3.3.
Preserve the notation above.(a) F ( k, B ) ⊆ F reg ( k, B ) ⊆ F ( k, B ) .(b) If | Max( B ) | ≤ q , then F ( A, B ) = F reg ( A, B ) .(c) If F ( A, B ) = F reg ( A, B ) , then the restriction map from Mult( k, B, F ( k, B )) to Mult( k, B, F ( k, B )) is an order isomorphism.Proof. (a) is obvious.(b) Let I ∈ F ( k, B ). If I is not regular, then I is contained in theunion of the maximal ideals of B ; by [8, Lemma 3.6], it follows that I is contained in some maximal ideal of B . But this would imply 1 / ∈ IB ,a contradiction.(c) Let I ∈ F ( k, B ). By hypothesis, I contains a regular element u of B , which is a unit since B is Artinian; in particular, u − I =( I : u ) ∈ F ( k, B ) ⊆ F ( k, B ). Hence, ( u − I ) ⋆ ⊆ u − I ⋆ for everymultiplicative operation ⋆ on ( k, B, F ( k, B )); since we can do thesame with the unit v := u − , it follows that ( u − I ) ⋆ = u − I ⋆ . Hence,every ⋆ ∈ Mult( k, B, F ( k, B )) is uniquely determined by its action SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 5 on F ( A, B ), and so the restriction map is injective; furthermore, ev-ery ♯ ∈ Mult(
A, B, F ( A, B )) can be extended to F ( A, B ) by setting I ⋆ := u ( u − I ) ⋆ when u ∈ I , and thus the map is also surjective. (cid:3) Note that the equality F ( k, B ) = F reg ( k, B ) may indeed fail: for ex-ample, if k = F and B = k the set V := { (0 , , , (1 , , , (1 , , , (0 , , } is a k -module in F ( A, B ), but it is not regular.
Definition 3.4. A star operation on ( k, B ) is a multiplicative opera-tion ⋆ on ( k, B, F ( A, B )) such that k ⋆ = k . We denote their set by Star( k, B ) . The terminology is justified by the following situation: let ( D, m )be a Noetherian local integral domain of dimension 1, and let T beits integral closure. Suppose that T is finite over D , and suppose that( D : T ) = m . By [19, Theorem 6.4 and Corollary 6.5], there is an order-preserving bijection between the set of star operations on D and the setof star operations on ( D/ m , T / m ): in particular, Star( D ) depends onlyon the extension k ⊆ T / m of Artinian rings, i.e., from the structureof the Artinian ring T / m as a k -algebra. Such a ring exists for everychoice of s , as the next proposition shows. Proposition 3.5.
Let s be a sequence of pairs of positive integers.Then, there is a local Noetherian domain D s of dimension with inte-gral closure T such that ( D s : T ) = m and T / m ≃ B s .Proof. Let L , . . . , L t be the residue fields of B s and, for every i , let f i ( X ) ∈ k [ X ] be an irreducible polynomial with splitting field L i . In thepolynomial ring k [ X , . . . , X t ], let P i be the prime ideal generated by f i ( X i ), and let S := k [ X , . . . , X t ] \ S i P i . Then, T := S − [ X , . . . , X t ]is a principal ideal domain with t maximal ideals, P T, . . . , P t T , suchthat the residue field of P i T is L i .Let I := P e · · · P e t t T . Then, T /I is isomorphic to B s ; if π is thequotient T −→ B s , defining D s := π − ( k ) we have the ring we werelooking for. (cid:3) Note that, while B s can be defined in a “canonical” way, there isa lot more freedom in defining D s , since it may be possible to use apolynomial ring with fewer indeterminates (for example, if each L i hasa different degree, then it is sufficient to consider T as a localization of k [ X ]). Definition 3.6.
Let s be a sequence of pairs of positive integers. Wedenote by Λ( s , q ) the cardinality of Star( k, B s ) , where q := | k | . If s is fixed, we can expect the function q Λ( s , q ) to be increasing,since as q grows the ring B s contains more and more subspaces andthus more and more closure operations. Indeed, this is exactly whathappens for n ( s ) >
3, while for n ( s ) ≤ s , q ) is constant DARIO SPIRITO (see [8] and the proof of Proposition 3.9 below). To study how fast itgrows, we introduce the following function.
Definition 3.7.
Let s be a sequence of pairs of positive integers. Weset θ ( s , q ) := log q log Λ( s , q ) . It is not at all obvious that this is a good choice. However, we havethe following.
Proposition 3.8.
For every n ∈ N , we have lim q →∞ θ (( n, , q ) = ( ( n − if n is even , ( n − n − if n is odd.Proof. If s = { ( n, } , the ring B := B s is just the degree n extensionof k . Let γ n be the number on the right hand side in the formula of thestatement.Suppose n is even. By [18, Theorem 3.7], for every ǫ > q , q γ n ≤ log | Star( k, B ) | ≤ (1 + ǫ ) q γ n ;taking the logarithm in base q we have γ n ≤ θ (( n, , q ) ≤ γ n + log q (1 + ǫ )and the claim follows by taking the limit. If n is odd, instead of 1 + ǫ we have 2 + ǫ , but the same reasoning applies. (cid:3) Another case when we can calculate the limit of θ ( s , q ) is for low n ( s ). Proposition 3.9.
Let s be a sequence of pairs of positive integers.Then:(1) if n ( s ) ≤ , then θ ( s , q ) = −∞ for every q ;(2) if n ( s ) = 3 , then lim q →∞ θ ( s , q ) = 0 Proof.
By Proposition 3.5, we can find a local Noetherian domain( D, m ) of dimension 1 such that Star( D ) ≃ Star( k, B ); let T be theintegral closure of D . Then, ℓ ( T /D ) = n ( s ).If ℓ ( T /D ) ≤
2, then by [2, Theorem 6.3] or [13, Theorem 3.8] Star( D )is a singleton, and thus θ ( s , q ) = log q log −∞ . If n ( s ) = 3, then by [8, Theorem 3.1] the cardinality of Star( D ) doesnot depend on k = D/ m . Hence, log (Λ( s , q )) is constant in q , and so θ ( s , q ) = log q log (Λ( s , q )) goes to 0. (cid:3) In view of these two cases, we advance the following conjectures.
SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 7
Conjecture 1.
For every sequence of pairs of positive integers s , thefunction q θ ( s , q ) has a limit when q → ∞ . Furthermore, Conjecture 2.
The limit of θ ( s , q ) as q → ∞ depends only on n ( s ),that is, if n ( s ) = n ( t ) then θ ( s , q ) and θ ( t , q ) have the same limit as q → ∞ .It is to be noted that Propositions 3.8 and 3.9 are quite flimsy ev-idence for these conjectures; in particular, Proposition 3.9 would alsobe true if, instead of the very peculiar function q log q log Λ( s , q ),we would have taken any function q ζ ( q ) with limit 0 as q → ∞ . Inthe rest of the paper, we will give some justification for this choice byshowing that, as q → ∞ , the limit inferior and superior of θ ( s , q ) canbe bounded by functions in n ( s ) with a polynomial growth (Theorem5.9) and that Conjecture 2 is true also for n = 4.4. The number of subspaces
Let A ⊆ B be a ring extension and let G ⊆ F B ( A ). Then, a mul-tiplicative operation ⋆ on ( A, B, G ) is uniquely determined by the set G ⋆ := { I ∈ G | I = I ⋆ } of its closed ideals. In particular, there is anobvious bound | Mult(
A, B, G ) | ≤ |G| . In this section, we use this factto bound the number of star operations on ( k, B ).Given a finite field k of cardinality q , and a k -vector space V ofdimension n , we denote by Z ( q, n ) the number of vector subspaces of V . Lemma 4.1.
For every n and every ǫ > , there is a q ( n, ǫ ) such that Z ( q, n ) ≤ ( n · q n / ǫ ( n/ if n is even ,n · q ( n − / ǫ (( n +1) / if n is oddfor all q ≥ q ( n, ǫ ) .Proof. Fix ǫ > Z t ( q, n ) be the number of t -dimensional subspaces of V . Then(see e.g. [21, Proposition 1.3.18] or [5, Chapter 13, Proposition 2.1]), Z t ( q, n ) is given by the q -binomial coefficient (cid:18) nt (cid:19) q := ( q n − · · · ( q n − t +1 − q t − · · · ( q − . For large q , we have q n − t + j − ≤ q n − t + ǫ ( q j − q n − t + j + ǫ . Hence, (cid:18) nt (cid:19) q = n − t Y j =1 q n − t + j − q j − ≤ t Y j =1 q n − t + ǫ = q t ( n − t + ǫ ) DARIO SPIRITO
The maximum of the function t t ( n − t + ǫ ) is in t := n + ǫ ; checkingthe integers closer to t , we obtain that the maximum is in n when n is even and in n +12 when n is odd.Therefore, if n is even we have Z ( q, n ) = n X t =0 Z t ( q, n ) ≤ n X t =1 q ( n/ n − ( n/ ǫ ) ≤ nq n / ǫn/ (where we can forget the zero subspace since the inequalities are nottight). Analogously, if n is odd, Z ( q, n ) = n X t =0 Z t ( q, n ) ≤ n X t =1 q (( n +1) / n − (( n +1) / ǫ ) ≤ nq ( n − / ǫ ( n +1) / . The claim is proved. (cid:3)
Theorem 4.2.
Let s be a sequence of pairs of positive integers and let n := n ( s ) . Then, lim sup q →∞ θ ( s , q ) ≤ ( n ( n − if n is even , ( n − if n is odd . Proof.
By parts (b) and (c) of Proposition 3.3, for large q any staroperation on ( k, B s ) is a multiplicative operation on ( k, B, F ( k, B )),and thus | Star( k, B s ) | ≤ |F ( k,B s ) | , i.e., θ ( s , q ) ≤ log q |F ( k, B s ) | . The number of k -subspaces of B s containing 1 is the number of k -subspaces of a vector space of dimension n −
1, i.e., |F ( k, B s ) | ≤ Z ( q, n − n even,lim sup q →∞ θ ( s , q ) ≤ lim sup q →∞ log q (( n − · q (( n − − / ǫ (( n − / ) ≤≤ lim sup q →∞ (cid:18) ( n − −
14 + ǫ n (cid:19) = n − n n ( n − n − ǫ > n odd gives lim sup q →∞ θ ( s , q ) ≤ ( n − . (cid:3) The two quadratic functions that bound lim sup q →∞ θ ( s , q ) are ratherclose to the quadratic functions that give the limit of θ (( n, , q ) inProposition 3.8: for example, if n is even, then the difference betweenthe upper bound and the limit of the case ( n,
1) is just n ( n − − ( n − = n − , which is linear instead of quadratic.5. Lower bounds
Definition 5.1.
Let I ∈ F ( k, B ) . The principal star operation gener-ated by I is the largest star operation that closes I ; we denote it by ∗ I .If A ⊆ F ( k, B ) , the star operation generated by A , denoted by ∗ A , is SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 9 the largest star operation that closes every element of G ; equivalently, ∗ A = inf {∗ I | I ∈ A} . The definition given above is slightly different from the definitiongiven in [19, Section 4] since we are imposing that ∗ I also closes k (as we want ∗ I to be a star operation and not only a multiplicativeoperation). However, the two definitions are actually very close. Proposition 5.2.
Let
I, J ∈ F ( k, B ) . Then, J ∗ I = ( k if J = k S { ( I : b ) | bJ ⊆ I } otherwise . Proof.
Let v ( I ) be the multiplicative operation generated by I on ( k, B, F ( k, B )),as in [19, Definition 4.2]. Then, ∗ I is just the infimum of v ( k ) and v ( I ),and thus J ∗ I = J v ( k ) ∩ J v ( I ) . By the proof of [19, Lemma 4.4], we have J v ( I ) = S { ( I : b ) | bJ ⊆ I } .On the other hand, if J = k then clearly J v ( k ) = k ; if J = k , then( k : J ) = 0, since if tJ ⊆ k then t ∈ k (since 1 ∈ J ) but sincedim k J > t = 0. Hence, again by [19, Lemma4.4], J v ( k ) = ( k : ( k : J )) = ( k : 0) = B. The claim is proved. (cid:3)
Let ∼ be the equivalence relation defined by I ∼ J if and only if ∗ I = ∗ J . If G ⊆ F ( k, B ), we denote by [ G ] the set of equivalence classescontaining at least one element of G . By definition, | Star( k, B ) | ≥| [ F ( k, B )] | ; however, this bound is too weak, since it is not exponen-tial in the cardinality of [ F ( k, B )], which is typically polynomial in q .Hence, we need a stronger situation. Lemma 5.3.
Let
G ⊆ F ( k, B ) be a subset such that, for every I ∈ G ,we have I ∗ G\{ I } = I. Then, | Star( k, B ) | ≥ |G| . Note that the hypothesis implies that no two distinct elements of G are equivalent under ∼ . Proof.
Let
A 6 = B two subsets of G . Then, without loss of generalitythere is an I ∈ A \ B : we have I ∗ A ⊆ I ∗ I = I , so that I ∗ A = I , while I ∗ B ⊇ I ∗ G\{ I } ) I by hypothesis and since B ⊆ G \ { I } . Therefore, ∗ A = ∗ B . It fol-lows that every subset of G generates a different star operation, and so | Star( k, B ) | ≥ |G| . (cid:3) We now want to apply this criterion. We start from the local case.
Proposition 5.4.
Let e, f be positive integers and let n := ef . Then, θ (( e, f ) , q ) ≥ n − . for every q .Proof. If n ≤
3, then θ (( e, f ) , q ) ≥ ≥ n − n ≥
4. By definition, θ (( e, f ) , q ) is equal to the number of star opera-tions on ( k, B ), where B := L [ X ] / ( X f ) for some extension L of k andsome f ≥ α ∈ B \ k , consider the submodule I ( α ) := h , α i . Everysuch submodule can be obtained in q − q ways (through all the β ∈ I ( α ) \ k ) and thus there are exactly ( q n − q ) / ( q − q ) = ( q n − − / ( q − I be the set of the ideals I ( α ) not containing X f − .We want to consider I ( α ) ∗ I ( β ) . By Proposition 5.2, we have I ( α ) ∗ I ( β ) = \ { ( I ( β ) : γ ) | I ( α ) ⊆ ( I ( β ) : γ ) } . Suppose I ( α ) ⊆ ( I ( β ) : γ ): then, if γ is a unit of B we have I ( α ) ⊆ γ − I ( β ), and the two ideals must be equal since they are both 2-dimensional k -subspaces of B . On the other hand, if γ is not a unitthen γX f − = 0 and thus X f − ∈ ( I ( β ) : γ ). Therefore, we have twocases: • if I ( α ) ⊆ ( I ( β ) : γ ) for some unit γ , then I ( α ) ∗ I ( β ) = I ( α ) = γ − I ( β ) and I ( α ) and I ( β ) are in the same class; • if I ( α ) * ( I ( β ) : γ ) for all units γ , then X f − ∈ I ( α ) ∗ I ( β ) .Note that, if f = 1, then B is a field and thus in the second case I ( α ) * ( I ( β ) : γ ) for all γ = 0, and thus I ( α ) ∗ I ( β ) = B .Let G be a subset of I containing exactly one I ( α ) for each classwith respect to ∼ . If I ∈ G , then X f − ∈ I ∗ J for every J ∈ G \ { I } ;thus, X f − ∈ I ∗ G\{ J } . Since X f − / ∈ I by definition, by Lemma 5.3 wehave | Star( k, B ) | ≥ |G| .We now need to estimate |G| . By the reasoning above, the ideals I ( α )and I ( β ) can be in the same class only if I ( β ) = γI ( α ) for some unit γ .Since 1 ∈ I ( α ), the ideal I ( β ) can be in this form only if γ − ∈ I ( β ), andthus there are at most q − γI ( α ) = tγI ( α )for all t ∈ k \ { } , and thus the number of such ideals is at most( q − / ( q −
1) = q + 1. Hence, |G| ≥ q + 1 (cid:18) q n − − q − − (cid:19) = q n − − qq − ≥ q n − . The claim follows. (cid:3)
The previous proof does not quite work in the non-local case, sincein this case B does not have an essential B -ideal analogous to X f . Afirst attempt is the following. SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 11
Proposition 5.5.
Let B , B two k -algebras. There is an injective,order-preserving map Star( k, B ) × Star( k, B ) ֒ → Star( k, B × B ) . In particular, | Star( k, B ) | ≤ | Star( k, B × B ) | .Proof. Let G i := F ( k, B i ). For each ⋆ ∈ Star( k, B ), let H := { I × B | I ∈ G ⋆ } and define symmetrically H . For every pair ( ⋆ , ⋆ ) inthe product Star( k, B ) × Star( k, B ), let ι ( ⋆ , ⋆ ) be the star operationgenerated by H ∪H : then, ι is a map from the product to Star( k, B × B ). To show that it is injective, let J := I × B for some I ∈ G . Then,for every L = L × B ∈ H , we have( J : L ) = ( I × B : L × B ) = ( I : L ) × B and thus J ∗ L = I ∗ L × B ; On the other hand, if L = B × L ∈ H ,then J ∗ L = B × B . Therefore, J ι ( ⋆ ,⋆ ) = I ⋆ × B ; in the same way,if J = B × I , then J ι ( ⋆ ,⋆ ) = B × I ⋆ . Hence, ι is injective and theclaim is proved. (cid:3) The previous proposition works best when B has a local factor B i such that n i = e i f i is large; in that case, using Proposition 5.4 have | Star( k, B ) | ≥ | Star( k, B i ) | ≥ q ni − , which gives θ ( s , q ) ≥ n i − n i is small with respect to n we need a different kindof estimate. Lemma 5.6.
Let B be a finite k -algebra, and let n := ℓ k ( B ) . Then, forevery ǫ > there is a q ( ǫ ) such that if q ≥ q ( ǫ ) the ring B s has at least ( q − n ≥ q (1 − ǫ ) n units.Proof. Let B := B × · · · × B t , where B i := F q ei [ X ] / ( X f i ) for each i .Set n i := e i f i . Each B i has q e i f i − q e i ≥ q n i − q n i − units; thus thenumber of units of B is at least t Y i =1 ( q n i − q n i − ) ≥ t Y i =1 q n i − ( q −
1) = q n − ( t − ( q − t ≥≥ ( q − n = q n log q ( q − ≥ q (1 − ǫ ) n for large q . The claim is proved. (cid:3) Proposition 5.7.
Let s := { ( e , f ) , . . . , ( e t , f t ) } be a sequence of pairsof positive integers, and let n := n ( s ) and r := e t f t . If n − r ≥ , then lim inf q →∞ θ ( s , q ) ≥ n − r − . Proof.
Since we are only interested in the limit for large q , we cansuppose that q > t , so that we only need to consider subspaces in F ( k, B ). Let B i := F q ei [ X ] / ( X f i ), write B := B × · · · × B t and B ′ := B ×· · · × B t − , so that B = B ′ × B t . By Lemma 5.6, for large q the ring B ′ has at least q (1 − ǫ )( n − r ) units. For every unit α of B ′ not belonging to k ,let I ( α ) := h , ( α, i , and let I be the set of such I ( α ).We note that the unique elements of I ( α ) in the form ( γ,
0) are thosewith γ = zα , for some z ∈ k : hence, the cardinality of I is equal to | U ( B ′ ) | q − ≥ q (1 − ǫ )( n − r ) − q − ≥ q (1 − ǫ )( n − r ) − for large q . Furthermore, the second component of any element of I ( α )belongs to k .By Proposition 5.2, we have I ( α ) ∗ I ( β ) = \ { ( I ( β ) : γ ) | γI ( α ) ⊆ I ( β ) } . We claim that if γ = 0 and γI ( α ) ⊆ I ( β ) then γ is a unit of B .Indeed, let γ := ( γ , γ ). Since γI ( α ) ⊆ I ( β ), we must have γ · ∈ I ( β ) and γ ( α, ∈ I ( β ). The second component of γ = γ · γ ,and thus must belong to k . If γ = 0, then γ ( α,
0) = ( γ α,
0) and thus γ α = sβ for some s ∈ k ; in particular, I ( α ) = I ( β ).If γ = 0 and γ = 0, then (0 , γ ) ∈ I ( β ); however, this would implythat (1 ,
0) = (1 , − γ − γ ∈ I ( β ), against α = k .Suppose γ , γ = 0. Then, γ α = sβ for some s ∈ k ; since α is a unit, sβ = 0 and thus sβ is a unit of B ′ , and thus the same must hold for γ α ; in particular, γ must be a unit of B ′ , and thus γ = ( γ , γ ) is aunit of B , as claimed.Therefore, if γ = 0 and γI ( α ) ⊆ I ( β ) then ( I ( β ) : γ ) = γ − I ( β )has dimension 2, and thus must be equal to I ( α ). Hence, I ( α ) ∗ I ( β ) iseither equal to I ( α ) or to B , and in the former case it must be in theform γ − I ( β ) for some unit γ of B belonging to I ( β ); in particular, I ( α ) and I ( β ) must be in the same class. Let G be a set constructedby taking one representative for each class. There are at most q − q possible units γ (for each choice of α ), and they give q different ideals γ − I ( β ), since γ − I ( β ) = ( sγ ) − I ( β ) for every s ∈ k . Hence, |G| ≥ q (1 − ǫ )( n − r ) − q = q (1 − ǫ )( n − r ) − for large q .As in the proof of Proposition 5.4, G satisfies the hypothesis ofLemma 5.3, and thus | Star( k, B ) | ≥ |G| . Using the previous estimateand taking the limit as q → ∞ we have our claim. (cid:3) Proposition 5.8.
Let s be a sequence of pair of positive integers, andlet n := n ( s ) . Then, lim inf q →∞ θ ( s , q ) ≥ ( n − if n is even , n − if n is odd . SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 13
Proof.
Let s := { ( e , f ) , . . . , ( e t , f t ) } , and let n i := e i f i for each i .Suppose first that n is even, and let r be the largest of the n i . If r ≤ n/
2, then by Proposition 5.7 we have, for large q ,lim inf q →∞ θ ( s , q ) ≥ n − n − n − n − . On the other hand, if r > n/ r ≥ ( n + 2) /
2, and thus by Propo-sitions 5.4 and 5.5 we have θ ( s , q ) ≥ r − ≥ n − n − , and the claim follows.Suppose now that n is odd. If t = 1 (i.e., if the associated Artinianring B is local) then by Proposition 5.4 we have θ ( s , q ) ≥ n − ≥ n − .If t >
1, then there is at least one n i that is smaller than n/
2; let it be r . Then, r ≤ ( n − / q →∞ θ ( s , q ) ≥ n − n − − n − . The claim is proved. (cid:3)
Theorem 5.9.
Let s be a sequence of pair of positive integers, and let n := n ( s ) . Then, n − ≤ lim inf q →∞ θ ( s , q ) ≤ lim sup q →∞ θ ( s , q ) ≤ n ( n − if n is even ,n − ≤ lim inf q →∞ θ ( s , q ) ≤ lim sup q →∞ θ ( s , q ) ≤ ( n − if n is odd . Proof.
The first inequality (of both cases) follows from Proposition 5.8,and last from Theorem 4.2. (cid:3)
Note that the limit inferior of the previous theorem is only significantfor n ≥
5. For n ≤ n = 4 in Section 7.6. Codimension 1
In this section, we consider the subspaces of B having codimension 1;in particular, we want to show that the number of classes of these ideals(with respect to generating the same star operation) can be boundedabove by a function that does not depend on the size of k .The main tool is the use of canonical ideals. Given an integral domain D with quotient field K , an ideal I of D is a canonical ideal of D if,for every ideal J , we have J = ( I : K ( I : K J )); note that there aremany equivalent (and more general) definitions (see e.g. [14, Chapter15]), but we use this one since it is closer to our subject. In particular,we need the following characterization. Proposition 6.1.
Let ( D, m ) be a one-dimensional Noetherian localdomain, let T be its integral closure, and suppose that ( D : T ) = m . Let I ∈ F ( D, T ) . Then, I is a canonical ideal if and only if ℓ D ( T /I ) = 1 and m is the largest ideal of T contained in I .Proof. By [14, Theorem 15.5], I is a canonical ideal if and only if ℓ D (( I : m ) /I ) = 1.Since m is a T -ideal contained in I , we have I ( T ⊆ ( I : m );hence, ℓ D (( I : m ) /I ) = 1 if and only if ( I : m ) = T and ℓ D ( T /I ) = 1.Furthermore, if L is a T -ideal contained in I , then ( I : m ) ⊇ ( L : m ),which is greater than T as soon as m ( L . Hence, m must be the largestideal of T contained in I .Conversely, if the two conditions hold, we claim that ( I : m ) = T :otherwise, there would be t ∈ ( I : m ) \ T , but then t m would be a T -ideal contained in I but not in m , against the hypothesis. Therefore ℓ D (( I : m ) /I ) = ℓ D ( T /I ) = 1 is a canonical ideal. (cid:3)
Suppose now we are in our setting: k is a field and B is a finite k -algebra. Following [19], we say that I ∈ F k ( B ) is a canonical ideal of ( k, B ) if ∗ I is the identity; this definition mirrors the definition of m -canonical ideals given in [6]. By Proposition 5.2 and the explicitdescription of the closure generated by I (see [19, Lemma 4.4]), for I ∈ F ( k, B ) this is equivalent to the condition that ( I : B ( I : B J )) = J for all J ∈ F ( k, B ) \ { k } . Proposition 6.2.
Let ( D, m ) be a one-dimensional Noetherian localdomain, let T be its integral closure and suppose ( D : T ) = m . Suppose I is a fractional ideal of D satisfying ( D : T ) ⊆ I ⊆ T . Then, I is acanonical ideal of D if and only if its image J is a canonical ideal of ( D/ m , T / m ) .Proof. By [19, Theorem 6.4 and Corollary 6.5] (see also the discussionafter Definition 3.4), there is a bijection between the star operationson D and the star operations on ( D/ m , T / m ); in particular, the staroperation generated by I correspond to the star operation generatedby its image J . The claim follows from the definitions. (cid:3) Corollary 6.3.
Let B be a finite k -algebra that is a principal ideal ring,and let I ∈ F ( k, B ) . Then, I is a canonical ideal of ( k, B, F ( A, B )) ifand only if ℓ k ( B/I ) = 1 and I does not contain any nonzero B -ideal.Proof. By Proposition 3.5, we can find a principal ideal domain T suchthat B is a quotient of T . Setting D to be the counterimage of k , we arein the setting of Proposition 6.2, and then we can apply Proposition6.1. (cid:3) Given a k -algebra B , let now H ( B ) be the set of all k -hyperplanes of B containing 1, i.e., the set of I ∈ F ( k, B ) such that ℓ k ( B/I ) = 1. We
SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 15 want to estimate in how many different star operations the elements of H ( B ) generate, i.e., in how many classes H ( B ) is partitioned.If I ∈ F ( A, B ), we define Z ( I ) to be the largest B -ideal contained in I : this is well-defined since if L , L ⊆ I are B -ideals then also L + L is a B -ideal contained in I . Lemma 6.4.
Let
I, J ∈ F ( A, B ) , and let ∗ I be the multiplicative op-eration generated by I . Then, J + Z ( I ) ⊆ J ∗ I .Proof. Since J ⊆ J ∗ I , we only need to show that Z ( I ) ⊆ J ∗ I . If t ∈ Z ( I ), then tT ⊆ Z ( I ) ⊆ I ; in particular, t ( I : B J ) ⊆ I and thus t ∈ ( I : B ( I : B J )) = J ∗ I (the last equality coming from [19, Lemma4.4] and Proposition 5.2. (cid:3) Proposition 6.5.
Let B be a finite k -algebra that is a principal idealring, and let n := ℓ k ( B ) . Then, the following hold.(a) If I, J ∈ H ( B ) , then ∗ I = ∗ J if and only if Z ( I ) = Z ( J ) .(b) If ℓ k ( B ) = n , then H ( B ) is divided into at most n classes.Proof. (a) If Z ( I ) = Z ( J ), then without loss of generality Z ( I ) * Z ( J )and thus Z ( I ) * J . By Lemma 6.4, B = J + Z ( I ) ⊆ J ∗ I , so that J = J ∗ I and I and J are not in the same class.Conversely, suppose Z ( I ) = Z ( J ), and consider B ′ := B/Z ( I ). Then, I/Z ( I ) and J/Z ( I ) are both hyperplanes of B ′ not containing any B ′ -ideal; by Corollary 6.2, they are canonical ideals of B ′ . Hence, theygenerate the identity star operation on B ′ , and so in B they generatethe map ∗ I : L L + Z ( I ). In particular, I and J are in the sameclass.(b) Let B = B ×· · ·× B t . The ideals of B are in the form I ×· · ·× I t ,where each I i is either B i or an ideal of B i . Since B i ≃ L i [ X ] / ( X f i )for some extension L i of k , there are f i + 1 possible I i , and thus thenumber of ideals of B is ( f + 1) · · · ( f t + 1). However, t Y i =1 ( f i + 1) ≤ (cid:18) ( f + 1) + · · · + ( f t + 1) t (cid:19) t ≤ (cid:18) n + tt (cid:19) t = (cid:16) nt (cid:17) t using the inequality between geometric and arithmetic mean and thefact that f + · · · + f t ≤ n . Furthermore, the function t (cid:0) nt (cid:1) t isincreasing in t , and thus, since t ≤ n , the number of classes is at most(1 + 1) n = 2 n , as claimed. (cid:3) Remark 6.6. (1) The bound of Proposition 6.5 does not depend on the size of k .(2) Not all ideals of B are equal to Z ( I ) for some ideal I of codi-mension 1: for example, B itself doesn’t. Another case are thesubspaces that are themselves hyperplanes: in that case, weshould have I = Z ( I ), which is impossible if 1 ∈ I . (3) If n = 3, then the bound of Proposition 6.5 gives | Star( k, B ) | ≤
65 for all choices of B . By considering in more details the differ-ent cases, and excluding the impossible B -ideals, it is possibleto obtain much tighter bounds; for example, if B = k thenwe have only three possible Z ( I ) = (0) (the ones in the form I × I × I with one I i equal to k and two I j equal to 0), andthus | Mult ( k, B ) | ≤ = 9, which is the exact value of | Star( k, B ) | by [8, Theorem 3.1(2)] and Proposition 3.9.7. The case n = 4In this section, we are going to prove Conjecture 2 for n ( s ) = 4. Inthis case, Theorem 5.9 gives0 ≤ lim inf q →∞ θ ( s , q ) ≤ lim sup q →∞ θ ( s , q ) ≤ , while Proposition 3.8 says that the limit of θ (( n, , q ) is 1. Hence, weneed to prove that θ ( s , q ) → s .We start from the upper bound. Lemma 7.1.
Let k be a finite field of cardinality q , let s := { ( e , f ) , . . . , ( e t , f t ) } be a sequence of pairs of positive integers and let B be the Artinianring associated to s . Let e := P i e i and n := ℓ k ( B ) = P i e i f i . Let Σ := { α ∈ B | α + rα + s = 0 for some r, s ∈ k } . Then, | Σ | ≤ q n − e +1 + 2 t ( q − q ) ≤ q n − t +1 + 2 t ( q − q ) Proof.
Let B = B × · · · × B t , where each B i is a local k -algebra, andlet n i := ℓ k ( B i ). Let π i : B i −→ L i be the quotient of B i on its residuefield L i .Let r, s ∈ k , and consider the equation f ( X ) = X + rX + s . Let S i ( f ) be the set of solutions of f ( X ) = 0 in B i , and suppose that S i = ∅ . If α ∈ S i , then we can factorize f ( X ) as ( X − α )( X − β ) forsome β ∈ B i . We distinguish two cases.If f ( X ) has simple roots in the algebraic closure k of k , then theimages of α and β in the residue field of B i are distinct. Hence, for every γ ∈ S i one of π i ( γ − α ) and π i ( γ − β ) is a unit, and thus ( γ − α )( γ − β ) = 0implies that γ = α or γ = β . Thus, | S i ( f ) | ≤ f ( X ) has a double root over k , then all roots of f ( X ) in B i mustbelong to π − i ( α ); since the latter is a coset of an ideal of B i , we have | S i ( f ) | ≤ q e i ( f i − = q n i − e i .Let now S ( f ) be the set of solutions of f ( X ) = 0 in B . Then, S ( f ) = S ( f ) × · · ·× S t ( f ); thus, if f ( X ) has simple roots in k then | S ( f ) | ≤ t ,while if f ( X ) has a double root then | S ( f ) | ≤ Q i q n i − e i = q n − e .There are exactly q polynomials of degree 2 with a double root (since k is finite, hence perfect), and thus q − q polynomials with single roots.Hence, | Σ | ≤ ( q − q ) · t + q · q n − e = q n − e +1 + 2 t ( q − q ) ≤ q n − t +1 + 2 t ( q − q ) , SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 17 as claimed. (cid:3)
Proposition 7.2.
Suppose n ( s ) = 4 . Then, lim sup q →∞ θ ( s , q ) ≤ . Proof.
Let B be the ring associated to s , and let G := F ( k, B ). Let G i := { I ∈ G | dim k I = i } . Then, G is empty, while G and G are bothsingletons: the former is { k } , the latter is { B } . Both are contained in G ⋆ for every star operation ⋆ on ( k, B ), and thus | Star( k, B ) | ≤ | [ G ] | + | [ G ] | .By Proposition 6.5, | [ G ] | ≤ G . Then, |G | = Z ( q,
3) = (cid:18) (cid:19) q = q − q − q + q + 1 . Let I be the set of the subspaces in the form h , α i , where α is a unitof B not belonging to k ; clearly, I ⊆ G . By Lemma 5.6, B has at least( q − units; furthermore, every I ( α ) contains at most q − q of them(not counting those in k ). Thus, |I| ≥ ( q − − qq − q = ( q − − q = q − q + 3 − q ≥ q − q + 3 . In particular, |G \ I| ≤ q + q + 1 − ( q − q + 3) = 4 q − α be a unit. Then, α − I ( α ) = α − h , α i = h , α − i = I ( α − ). Wehave that I ( α ) = I ( α − ) if and only if α − ∈ h , α i , that is, if and onlyif α satisfies an equation of degree 2 with coefficients in k . Since everyelement of I ( α ) is in the form s + tα for some s, t ∈ k , we have that α − I ( α ) = I ( α ) if and only if β − I ( α ) = I ( α ) for some unit β ∈ I ( α ).Therefore, I can be partitioned into two classes: • I := { I ( α ) | α − I ( α ) = I ( α ) } ; • I := { I ( α ) | α − I ( α ) = I ( α ) } ;Each element of I contains at least q − q units not in k ( q ele-ments, minus q in k , minus at most q that are not units). Thus, for eachsubspace there are at least ( q − q ) / ( q −
1) = q − q − ≥ q − | [ I ] | ≤ q + q + 1 − ( q + 16) q − ≤ q + 4 . Each subspace in I is equivalent only to himself; to estimate itscardinality, we distinguish two cases.If t ≥
2, by Lemma 7.1 there are at most q n − t +1 + 2 t ( q − q ) elementsthat are solutions of an equation of degree 2; hence, they can fill atmost q n − t +1 + 2 t ( q − q ) q − q = q n − t + 2 t ( q − q − ≥ q n − t − + 2 t ≥ q + 16 ideals, i.e., |I | ≤ q + 16. Therefore, | Star( k, B ) | ≤ [ G ]+[ G \I ]+[ I ]+[ I ] ≤ q +2+ q +16+ q +4 = 2 q +26 and lim sup q →∞ θ ( s , q ) ≤ log q log (2 q +26 ) = log q (6 q + 26) = 1 , as claimed.Suppose t = 1. Then, s = ( e, f ) with ef = 4, and thus we havefour possibilities: (4 , , , A := k [[ X , X , X , X ]],for which | Star( A ) | = 2 q +1 + 2 q +1 + 2 (see [22]), and thus the limit of θ ( s , q ) is again 1.The case s = { (2 , } correspond to B = L [ X ] / ( X ), where L isthe field of cardinality q . By Lemma 7.1, we have that the numberof solutions to equations of degree 2 is at most q − + 2( q − q ) = q + 2( q − q ). Reasoning as above, we get |I | ≤ q + 5, and so | Star( k, B ) | ≤ q q +2+ q +5+ q +4 = 2 q +15 . Hence, lim sup q θ ( s , q ) ≤
1, as claimed. (cid:3)
To study the limit inferior, we need to reason by cases. Let s = { ( e , f ) , . . . , ( e t , f t ) } : if n i := e i f i , then the set { n , . . . , n t } is a par-tition of n := n ( s ). In particular, if n = 4 then we have five possiblepartitions: • • • • • B . For example, the case 3 + 1 is further subdivided into thecases s = { (3 , , (1 , } (corresponding to B = L × k , where L is anextension of k of degree 3) and s = { (1 , , (1 , } (corresponding to B = k [ X ] / ( X ) × k ). Proposition 7.3.
Preserve the notation above. If n = 4 , then lim inf q θ ( s , q ) ≥ .Proof. Apply Proposition 5.4: s = { ( e, f ) } and thus θ (( e, f ) , q ) ≥ − (cid:3) Proposition 7.4.
Preserve the notation above. If e i = f i = 1 for some i , then lim inf q θ ( s , q ) ≥ .Proof. We can apply Proposition 5.7 with r = 1, obtaining lim inf q θ ( s , q ) ≥ − − (cid:3) SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 19
The only case remaining is the partition 2 + 2. To examine it, weapply a variant of the proof of Proposition 5.4.
Proposition 7.5.
Suppose s = { ( e , f ) , ( e , f ) } with e f = e f = 2 .Then, lim inf q →∞ θ ( s , q ) ≥ . Proof.
Let B = B × B be the ring associated to s .Suppose first that B = L is the extension of degree 2 of k . Then, B has q − q units not belonging to k , while B has at least q − q of them: hence, B has at least q ( q − q −
2) of these unit. For everysuch unit α of B , let I ( α ) := h , α i , and let I be the set of all I ( α ).No element of I ( α ) (different from 0) has a component equal to 0:otherwise, setting α = ( α , α ), we would have s + rα i = 0 for some s, r ∈ k not both 0, against the fact that α i is not in k . Therefore, eachthere are q − q elements β such that I ( α ) = I ( β ), and so I has atleast q ( q − q − q − q = q ( q −
2) elements.Let γ = ( γ , γ ) ∈ I ( α ): we claim that J := ( I ( α ) : γ ) cannot havedimension 3 over k .If γ = 0, then (0 , γ α ) ∈ I ( α ): then we must have γ α = 0, andsince α is a unit, this implies that γ = 0, i.e., that γ = 0 and thus J = B . If γ = 0, then γ is a unit of L . If dim k J >
2, then, J ∩ ( B × { } )must be nontrivial, i.e., there would be some β := ( β , = 0 such that β ∈ J . However, βγ = ( β , γ , γ ) = ( β γ , = (0 ,
0) since γ is aunit and β = 0. Hence, if γ = 0 then dim k J = 2, while if γ = 0 then J = B and dim k J = 4.Hence, I ( α ) ∗ I ( β ) is equal either to I ( α ) (if γ − I ( α ) = I ( β ) for someunit γ ∈ I ( α )) or to B ; in the former case I ( α ) and I ( β ) are in thesame class, in the latter they belong to different classes. As in theproof of Proposition 5.4, we have q − γ in I ( α ), and if t ∈ k \ { } then ( tγ ) − I ( α ) = γ − I ( α ), and thus each class contains atmost ( q − / ( q −
1) = q + 1 ideals of this kind.Let G be a set of representatives of the classes of I : then,[ G ] ≥ |I| q + 1 ≥ q ( q − q + 1 ≥ q − . By Lemma 5.3, it follows that | Star( k, B ) | ≥ q − , and thus lim inf q θ ( s , q ) ≥ lim inf q log q ( q −
3) = 1, as claimed.Clearly, if B = L then the argument is identical. Suppose thus that B = L = B : then, B and B must be isomorphic to k [ X ] / ( X ).Each B i has q − q units not belonging to k ; we choose those units α = ( α , α ) such that the image of α and α in k are distinct. Thus,we obtain ( q − q )( q − q ) = q ( q − q −
3) possible α . The rest ofthe reasoning proceeds in the same way, obtaining a bound |G| ≥ q − c for some constant c , from which the claim follows. (cid:3) Theorem 7.6.
Let s be a sequence of pairs of positive integers with n := n ( s ) = 4 . Then, lim q →∞ θ ( s , q ) = 1 . Proof.
By Proposition 7.2, lim sup q θ ( s , q ) ≤
1. By Propositions 7.3, 7.4and 7.5 (and the discussion before Proposition 7.3), on the other hand,lim sup q θ ( s , q ) ≥
1. Hence, the limit exists and is equal to 1. (cid:3)
The previous theorem does not allow by itself an explicit determi-nation of the number of star operations on ( k, B ), mainly because forevery s it is necessary to consider a different number of special cases,which give a different counting. (However, if one follows the proofs ina special case it is possible to obtain some more explicit estimates.) Intwo cases, the cardinality of Star( k, B ) has been determined explicitly: • if s = { (4 , } , then B = F (where F is the extension of k of degree 4) and the star operations on ( k, B ) correspond tothe star operations on the pseudo-valuation domain A := k + XF [[ X ]]; in this case, | Star( A ) | = 2 q +1 + 1by the results in [15] and [18]; • if s = { (1 , } then B = k [ X ] / ( X ), or in domain terms we arein the residually rational case A := k [[ X , X , X , X ]]; in thiscase, | Star( A ) | = 2 q +1 + 2 q +1 + 2by [22].We now show how to obtain a precise counting in a further specialcase, namely when B = L × k and L is the extension of k of degree 3. Example 7.7.
Let B := L × k , and let G and G be, respectively, theset of k -subspaces of B having dimension 2 and 3 over k that contain1. Let G := F ( k, B ) = F ( k, B ) (in this case t = 2 and thus the twosets are equal). On the set [ G ] of classes, define [ I ] (cid:22) [ J ] if and only if ∗ I ≥ ∗ J , that is, if and only if I = I ∗ J . Then, (cid:22) is a partial order on G ,and the set [ G ] ⋆ := { [ I ] | I ⋆ = I } is a downset for every ⋆ ∈ Star( k, B ).See [19, Definition 4.7 onwards] for more information about this order.As in the proof of Proposition 7.2, let G i be the set of I ∈ F ( k, B )having dimension i over k .By Proposition 6.5(a), the class of I ∈ G depends only on the largest B -ideal Z ( I ) contained into I . There are four ideals of B : the zero ideal, { } × B , B × { } and B × B . In particular, the latter two cannot bein the form Z ( I ), since B = B × B has dimension 4, while B × { } has dimension 3 and does not contain 1. Hence, we have two classes: • G , can := { I | Z ( I ) = { (0 , }} : these are the canonical ideals; • G , := { I | Z ( I ) = (0) × B } . SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 21
In particular, the elements of G , can close every k -subspace, and thus G , can is the maximum of ([ G ] , (cid:22) ). On the other hand, the elements of G , close exactly the subspaces J such that ((0) × B ) J ⊆ J , and theonly subspace satisfying this condition is R (1) := h (1 , , (1 , i = k × k .Note that R (1) does not close any other subspace different from k ,because there is no other subspace containing (0) × B . Thus, we havethe following partial graph of [ G ]: G , can → G , → [ R (1)] → [ k ] . Consider now I ∈ G . Then, we can write I = h (1 , , ( α, β ) i for some( α, β ) ∈ B . Since β ∈ k , we can suppose without loss of generality that β = 0, i.e., that I = R ( α ) := h (1 , , ( α, i . If α ∈ k then we have R ( α ) = R (1) (see above) and thus it is enough to consider the case α / ∈ k .There are q − q of such α , and each subspace can be generatedby q − q + q ideals in this form. Bythe proof of Proposition 5.7, every equivalence class of them contains q subspaces, and they are not comparable under (cid:22) . Furthermore, bydirect calculation ( I ( α ) : ( α, R (1) (so [ R (1)] (cid:22) [ I ( α )]) for every α , while no element of G , is closed by I ( α ). Thus, we obtain thefollowing complete picture of [ G ]: G , [ R (1)] [ A ] G , can [ R ( α )]...[ R ( α q +1 )]Hence, there are 2 q +2 + 2 nonempty downsets: • the whole [ G ]; • q +2 − X ∪ { [ A ] , R [1] } , for nonempty X ⊆ {G , , R ( α ) , . . . , R ( α q +1 ) } ; • [ A ] and [ A ] ∪ [ R (1)].By direct inspection and by [19, Proposition 4.10], all these downsetgive rise to a star operation, and thus | Star( k, B ) | = 2 q +2 + 2 staroperations.We note that, in the three cases considered explicitly for n = 4, thenumber of star operations is a polynomial f ( X ) evaluated in X = 2 q :indeed, for s = { (4 , } the polynomial is f ( X ) = 2 X + 1, for s = { (1 , } we have f ( X ) = 2 X + 2 X + 2, and for s = { (3 , , (1 , } wehave f ( X ) = 4 X + 2. This lead to the following, final Conjecture 3.
For every s such that n ( s ) = 4 there is a polynomial f ( X ) ∈ Z [ X ] such that Λ( s , q ) = f (2 q ) for every q .This conjecture cannot be generalized to higher lengths: for example,in the case s = { (5 , } , the number of star operations is equal to( q + 5)2 q − ( q −
1) [18, Theorem 4.3].
References [1] D. D. Anderson. GCD domains, Gauss’ lemma, and contents of polynomials.In
Non-Noetherian commutative ring theory , volume 520 of
Math. Appl. , pages1–31. Kluwer Acad. Publ., Dordrecht, 2000.[2] Hyman Bass. On the ubiquity of Gorenstein rings.
Math. Z. , 82:8–28, 1963.[3] Marco Fontana and Mi Hee Park. Star operations and pullbacks.
J. Algebra ,274(1):387–421, 2004.[4] Robert Gilmer.
Multiplicative Ideal Theory . Marcel Dekker Inc., New York,1972. Pure and Applied Mathematics, No. 12.[5] R. L. Graham, M. Gr¨otschel, and L. Lov´asz, editors.
Handbook of Combina-torics. Vol. 1, 2 . Elsevier Science B.V., Amsterdam; MIT Press, Cambridge,MA, 1995.[6] William J. Heinzer, James A. Huckaba, and Ira J. Papick. m -canonical idealsin integral domains. Comm. Algebra , 26(9):3021–3043, 1998.[7] Evan G. Houston, Abdeslam Mimouni, and Mi Hee Park. Integral domainswhich admit at most two star operations.
Comm. Algebra , 39(5):1907–1921,2011.[8] Evan G. Houston, Abdeslam Mimouni, and Mi Hee Park. Noetherian domainswhich admit only finitely many star operations.
J. Algebra , 366:78–93, 2012.[9] Evan G. Houston, Abdeslam Mimouni, and Mi Hee Park. Integrally closeddomains with only finitely many star operations.
Comm. Algebra , 42(12):5264–5286, 2014.[10] Evan G. Houston and Mi Hee Park. A characterization of local Noetheriandomains which admit only finitely many star operations: The infinite residuefield case.
J. Algebra , 407:105–134, 2014.[11] Thomas W. Hungerford. On the structure of principal ideal rings.
Pacific J.Math. , 25:543–547, 1968.[12] Wolfgang Krull. Beitr¨age zur Arithmetik kommutativer Integrit¨atsbereiche i-ii.
Math. Z. , 41(1):545–577; 665–679, 1936.[13] Eben Matlis. Reflexive domains.
J. Algebra , 8:1–33, 1968.[14] Eben Matlis. 1 -dimensional Cohen-Macaulay rings . Lecture Notes in Mathe-matics, Vol. 327. Springer-Verlag, Berlin-New York, 1973.[15] Mi Hee Park. On the cardinality of star operations on a pseudo-valuationdomain.
Rocky Mountain J. Math. , 42(6):1939–1951, 2012.[16] Pierre Samuel. Sur les anneaux factoriels.
Bull. Soc. Math. France , 89:155–173,1961.[17] Dario Spirito. Star operations on Kunz domains.
Int. Electron. J. Algebra ,25:171–185, 2019.[18] Dario Spirito. Vector subspaces of finite fields and star operations on pseudo-valuation domains.
Finite Fields Appl. , 56:17–30, 2019.[19] Dario Spirito. Multiplicative closure operations on ring extensions.
J. PureAppl. Algebra , to appear.[20] Dario Spirito. The sets of star and semistar operations on semilocal Pr¨uferdomains.
J. Commut. Algebra , to appear.
SYMPTOTIC FOR THE NUMBER OF STAR OPERATIONS 23 [21] Richard P. Stanley.
Enumerative combinatorics. Volume 1 , volume 49 of
Cam-bridge Studies in Advanced Mathematics . Cambridge University Press, Cam-bridge, second edition, 2012.[22] Bryan White.
Star Operations and Numerical Semigroup Rings . PhD thesis,The University of New Mexico, 2014.
Dipartimento di Matematica e Fisica, Universit`a degli Studi “RomaTre”, Roma, Italy
Current address : Dipartimento di Matematica, Universit`a di Padova, Padova,Italy
E-mail address ::