Star operations on Kunz domains

We study star operations on Kunz domains, a class of analytically irreducible, residually rational domains associated to pseudo-symmetric numerical semigroups, and we use them to refute a conjecture of Houston, Mimouni and Park. We also find an estimate for the number of star operations in a particular case, and a precise counting in a sub-case.


Introduction
Let D be an integral domain with quotient field K, and let F (D) be the set of fractional ideals of D, i.e., the set of D-submodules I of K such that xI ⊆ D for some x ∈ K \ {0}.
A star operation on D is a map ⋆ : F (D) −→ F (D), I → I ⋆ , such that, for every I, J ∈ F (D) and every x ∈ K: • I ⊆ I ⋆ ; • if I ⊆ J, then I ⋆ ⊆ J ⋆ ; • (I ⋆ ) ⋆ = I ⋆ ; • x · I ⋆ = (xI) ⋆ ; The easiest example of a non-trivial star operation is the v-operation v : I → (D : (D : I)), where if I, J ∈ F (D) we define (I : J) := {x ∈ K | xJ ⊆ I}. An ideal that is v-closed is said to be divisorial ; if I is divisorial and ⋆ is any other star operation then I = I ⋆ . We denote by d the identity, which is obviously a star operation.
Recently, the cardinality of the set Star(D) of the star operations on D has been studied, especially in the case of Noetherian [6,8] and Prüfer domains [5,7]. In particular, Houston, Mimouni and Park started studying the relationship between the cardinality of Star(D) and the cardinality of Star(T ), where T is an overring of D (an overring of D is a ring comprised between D and K) [3,4]: they called a domain star regular if |Star(D)| ≥ |Star(T )| for every overring of T . While even simple domains may fail to be star regular (for example, there are domains with just one star operations having an overring with infinitely many star operations [3, Example 1.3]), they conjectured that every one-dimensional local Noetherian domain D such that 1 < |Star(D)| < ∞ is star regular, and proved it when the residue field of D is infinite [3,Corollary 1.18].
In ∈ S}, where S is a numerical semigroup (i.e., a submonoid S ⊆ N such that N \ S is finite). Star operations can also be defined on numerical semigroups [13], and there is a link between star operations on S and star operations on being Gorenstein [2,10]. A detailed study of star operations on some numerical semigroup rings was carried out in [14].
In this paper, we study of star operations on Kunz domains, which are, roughly speaking, a generalization of rings in the form K[[S]] where S is a pseudo-symmetric semigroup (see the beginning of the next section for the definitions). We show that, if R is a Kunz domain whose residue field is finite and the length of R/R is at least 4 (where R is the integral closure of R) then R is a counterexample to Houston-Mimouni-Park's conjecture; that is, R satisfies 1 < |Star(R)| < ∞ but there is an overring T of R with more star operations than R. In Section 3, we also study more deeply one specific class of domains, linking the cardinality of Star(R) with the set of vector subspaces of a vector space over the residue field of R, and calculate the cardinality of Star(R) when the value semigroup of R is 4, 5, 7 .
We refer to [12] for information about numerical semigroup, and to [1] for the passage from numerical semigroup to one-dimensional local domains.
Let (V, M V ) be a discrete valuation ring with associated valuation v. Let (R, M R ) be a local subring of V with the following properties: • R and V have the same quotient field; • the integral closure of R is V ; • R is Noetherian; • the conductor ideal (R : V ) is nonzero; • the inclusion R ֒→ V induces an isomorphism of residue fields R/M R −→ V /M V . Equivalently, let R be an analytically irreducible, residually rational one-dimensional Noetherian local domain having integral closure V . Note that for every such R the set v(R) := {v(r) | r ∈ R} is a numerical semigroup. We state explicitly a property which we will be using many times.
Proposition 2.1 ([11, Corollary to Proposition 1]). Let R be as above, and let I ⊆ J be R-submodules of the quotient field of R. Then, where ℓ R is the length of an R-module.
We say that R is a Kunz domain if v(R) is a pseudo-symmetric semigroup [1, Proposition II. 1.12].
From now on, we suppose that R is a Kunz domain, and we set g := g(v(R)) and τ := g/2. The hypotheses on R guarantee that, if x ∈ V is such that v(x) > g, then x ∈ R [10, Theorem, p.749]. (a) T contains all elements of valuation g; Proof. Let y ′ ∈ V be another element of valuation g. Then, v(y/y ′ ) = 0, and thus c := y/y ′ is a unit of V . Hence, there is a c ′ ∈ R such that the images of c and c ′ in the residue field of V coincide; in particular, c = c ′ + m for some m ∈ M V . Hence, Since c ′ ∈ R, we have c ′ y ∈ R[y]; furthermore, v(my) = v(m) + v(y) > v(y) = g, and thus my ∈ R. Hence, y ′ ∈ R[y], and thus R[y] contains all elements of valuation g.
The last point follows from the fact that R + yR is an Rmodule, from R R + yR ⊆ T and from ℓ R (T /R) = 1.
In particular, the previous proposition shows that T is independent from the element y chosen. From now on, we will use T to denote this ring.
We denote by F 0 (R) the set of R-fractional ideals I such that R ⊆ I ⊆ V . If I is any fractional ideal over R, and α ∈ I is an element of minimal valuation, then α −1 I ∈ F 0 (R); hence, the action of any star operation is uniquely determined by its action on F 0 (R). Furthermore, V ⋆ = V for all ⋆ ∈ Star(R) (since (R : (R : V )) = V ) and thus I ⋆ ∈ F 0 (R) for all I ∈ F 0 (R), i.e., ⋆ restricts to a map from F 0 (R) to itself.
To analyze star operations, we want to subdivide them according to whether they close T or not. One case is very simple.
Proof. Suppose ⋆ = v: then, there is a fractional ideal I ∈ F 0 (R) that is ⋆-closed but not divisorial. By [1, Lemma II. 1.22], v(I) is not divisorial (in v(R)) and thus by [1, Proposition I.1.16] there is an integer n ∈ v(I) such that n + τ / ∈ v(I). Let x ∈ I be an element of valuation n, and consider the ideal Since v(x) > 0, every element of valuation g belongs to J; on the other hand, by the choice of n, no element of valuation τ can belong to J.
Consider now the ideal L := (R : We claim that T = J ∩ L: indeed, clearly J ∩ L contains R, and if y has valuation g then y ∈ J ∩ L by construction; thus Hence, T = J ∩L; since J and L are both ⋆-closed, so is T . Therefore, if T = T ⋆ then ⋆ must be the divisorial closure, as claimed. Suppose now that T = T ⋆ . Then, ⋆ restricts to a star operation ⋆ 1 := ⋆| F (T ) ; the amount of information we lose in the passage from ⋆ to ⋆ 1 depends on the R-fractional ideals that are not ideals over T . We can determine them explicitly.
Lemma 2.4. Let I ∈ F 0 (R), I = R. Then, the following are equivalent.
(ii) =⇒ (iii): since R ⊆ I, there is an element x of I of valuation 0; hence, IT contains an element of valuation g, and thus IT = I.
; hence, I contains an element (explicitly, xy) of valuation g and, by the proof of Lemma 2.2, it follows that it contains every element of valuation g.
Fix now an element y ∈ V of valuation g. Since IT = I, there are i ∈ I, t ∈ T such that it / ∈ I. By Lemma 2.2, there are r, r ′ ∈ R such that t = r + yr ′ ; hence, it = i(r + yr ′ ) = ir + iyr ′ . Both ir and iyr ′ are in I, the former since it belongs to IR = I and the latter because its valuation is at least g. However, this contradicts it / If v(I) = v(R), then we must have I = R, against our hypothesis; (i) ⇐⇒ (iv): by [9,Satz 5], I is the canonical ideal of R if and only if v(I) is the canonical ideal of v(R). The claim follows since v(R) is pseudo-symmetric and since the canonical ideal of a numerical semigroup S is S ∪{x ∈ N | g(S)−x / ∈ S}, which in this case is S ∪{τ }. For the last claim, we first note that (R : M R ) is divisorial (since M R is divisorial) and that is contains I: indeed, if x ∈ I has valuation τ , and m ∈ M R , then xm ∈ M R , for otherwise m / ∈ R and thus R + mR would be an ideal properly between R and I, against ℓ R (I/R) = 1. Hence, I v can only be I or (R : M R ). However, (R : I) ⊆ M R , and thus is well-defined and injective.
Proof. By Proposition 2.3, if ⋆ = v then T = T ⋆ , and thus ⋆| F (T ) is a star operation on T ; hence, Ψ is well-defined. We claim that it is injective: suppose ⋆ 1 = ⋆ 2 . Then, there is an I ∈ F 0 (R) such that By Lemma 2.4, I can only be R or a canonical ideal of R. In the former case, R ⋆ 1 = R = R ⋆ 2 , a contradiction. In the latter case, I ⋆ i can only if I or (R : M R ) (since ℓ((R : M R )/I) = 1); suppose now that I ⋆ = I for some ⋆ ∈ Star(R). By definition of the canonical ideal, J = (I : (I : J)) for every ideal J; since (I : An immediate corollary of the previous proposition is that |Star(R)| ≤ |Star(T )| + 2. Our counterexample thus involves finding star operations of T that do not belong to the image of Ψ; to do so, we restrict to the case ℓ R (V /R) ≥ 4 or, equivalently, |N \ v(R)| ≥ 4. This excludes exactly two pseudo-symmetric numerical semigroups, namely 3, 4, 5 and 3, 5, 7 .
Lemma 2.6. Let S be a pseudo-symmetric numerical semigroup, let g := max(N \ S) and let S ′ : Proof. We claim that a := τ and b := g − µ are the two elements we are looking for.
Since a + M S ⊆ S and a + g > g (and so a + g ∈ M S ) we have a ∈ (S ′ − M S ′ ). Furthermore, since |N \ S| ≥ 4, we have g > µ, and thus b + m ≥ g for all m ∈ M S ′ .
By the previous point, a + m, b + m ∈ S ′ ∪ {a, b} for every m ∈ M S ′ . We always have 2a ≥ g, and thus 2a ∈ S ′ .
Proposition 2.7. Let K be the residue field of R, and suppose that ℓ R (V /R) ≥ 4. There are at least |K| + 1 star operations on T that do not close (R : M R ).
Proof. We first note that (R : M R ) is a T -module. Indeed, let x ∈ (R : M R ) and t ∈ T : then, t = r + ay, with r ∈ R and v(y) = g, and so xt = xr+axy. Both xr and axy belong to (R : M R ), the former because (R : M R ) is a R-module and the latter since its valuation is at least g: hence, xt ∈ (R : M R ). Thus, it makes sense to ask if a star operation on T closes (R : M R ). We also note that T We claim that ⋆ i closes T i but not T j for j = i.
Indeed, clearly T ⋆ i i = T i . If j = i, then T i T j contains both x + α i y and x + α j y, and thus it contains their difference (α i − α j )y. Since α i and α j are units corresponding to different residues, it follows that α i − α j is a unit of R, and thus of T ; hence, y ∈ T i T j . By construction, y ∈ (T : M T ): thus, y ∈ T contains y, while y / ∈ (R : M R ). To conclude the proof, it is enough to note that none of the ⋆ i are the divisorial closure (since they close one of the T i , none of which are divisorial), and thus we have another star operation that does not close (R : M R ).
We are now ready to show that R is the desired counterexample.
Theorem 2.8. Let R be a Kunz domain with finite residue field, and suppose that ℓ R (V /R) ≥ 4. Then, 1 < |Star(R)| < ∞, but R is not star regular.
We note that this semigroup is pseudo-symmetric also if n = 3, for which the number of star operations has been calculated in [8, Proposition 2.10]: we have |Star(R)| = 4.
By Lemma 2.4, the only I ∈ F 0 (R) such that IT = I are R and the canonical ideals. From now on, we denote by G the set {I ∈ F 0 (R) | IT = I}; we want to parametrize G by subspaces of a vector space. Proof. Every I ∈ G contains T . The quotient of R-modules π : V → V /T induces a map π : G −→ P(V /T ) where P(V /T ) denotes the power set of V /T . It is obvious that π is injective.
The map π induces on V /T a structure of K-vector space of dimension n − 1. If I ∈ G, then its image along π will be a vector subspace; conversely, if W is a vector subspace of V /T then π −1 (W ) will be an ideal in G. The claim is proved.
For an arbitrary domain D and a fractional ideal I of D, the star operation generated by I is the map [13, Section 5] this star operation has the property that, if I is ⋆-closed for some ⋆ ∈ Star(R) and J is ⋆ I -closed, then J is also ⋆-closed. If ∆ ⊆ F (S), we define ⋆ ∆ as the map In the present case, we can characterize when an ideal is ⋆ ∆ -closed. (c) Suppose ; hence, using the previous point, Conversely, suppose I = I ⋆ J . Since I is nondivisorial, there must be γ ∈ (I : J), γ = 0 such that I ⊆ γ −1 J and I v γ −1 J. If v(γ) > 0, then γ −1 J contains the elements of valuation n−1; it follows that I v ⊆ γ −1 J and thus that I v ⊆ I ⋆ J , against I = I ⋆ J . Hence, v(γ) = 0, as claimed.
Conversely, suppose I = I ⋆ ∆ . For every J ∈ ∆, the ideal I ⋆ J is contained in I v = I ∪ {x | v(x) ≥ n − 1}; since ℓ(I v /I) = 1, it follows that I ⋆ J is either I or I v . Since I = I ⋆ ∆ , it must be I ⋆ J = I for some J; by the previous point, I ⊆ γ −1 J for some γ, as claimed.
An important consequence of the previous proposition is the following: suppose that ∆ is a set of nondivisorial ideals in F 0 (R) such that, when I = J are in ∆, then I γ −1 J for all γ having valuation 0. Then, for every subset Λ ⊆ ∆, the set of ideals of ∆ that are ⋆ Λ -closed is exactly Λ; in particular, each nonempty subset of ∆ generates a different star operation.
We will use this observation to estimate the cardinality of Star(R) when the residue field is finite. Proposition 3.3. Let R be a Kunz domain such that v(R) = n, n + 1, . . . , 2n − 3, 2n − 1 , and suppose that the residue field of R has cardinality q < ∞.Then, Let e 1 be an element of valuation 1, and let e i := e i 1 ; then, {1 = e 0 , e 1 , . . . , e n−1 } projects to a K-basis of A, which for simplicity we still denote by {e 0 , . . . , e n−1 }. The vector subspace spanned by e 0 is exactly the field K.
Since V and L are stable by multiplication by every element of valuation 0, asking if γI ⊆ J for some I, J ∈ F 0 (R) and some γ is equivalent to asking if there is a γ ∈ A of "valuation" 0 such that γI ⊆ J, where I and J are the images of I and J, respectively, in A. Hence, instead of working with ideals in F 0 (R) we can work with vector subspaces of A containing e 0 . Furthermore, if V is a vector subspace of A and γ has valuation 0, then γV has the same dimension of V ; thus, if V and W have the same dimension, γV ⊆ W if and only if γV = W . Let ∼ denote the equivalence relation such that V ∼ W if and only if γV = W for some γ of valuation 0.
Let X be the set of 2-dimensional subspaces of A that contain e 0 but not e n−1 . Then, the preimage of every element of X is a nondivisorial ideal.
Let V ∈ X, say V = e 0 , f , and consider the equivalence class ∆ of V with respect to ∼. Then, W ∈ ∆ if and only if γW = V for some γ; since 1 ∈ W , it follows that such a γ must belong to V . Since γ has valuation 0, it must be in the form λ 0 e 0 +λ 1 f with λ 0 = 0; furthermore, if γ ′ = λγ then γ −1 V = γ ′−1 W . Hence, the cardinality of ∆ is at most Therefore, X contains elements belonging to at least 1 q equivalence classes; let X ′ be a set of representatives of such classes, and let Y be the preimage of X ′ in the power set of F 0 (R). Then, every subset of Y generates a different star operation (with the empty set corresponding to the v-operation); it follows that as claimed.
Proof. Consider the same setup of the previous proof. We start by claiming that two vector subspaces W 1 , W 2 of A of dimension 3 that contain e 0 but not e 3 are equivalent under ∼.
Consider now the set ∆ of nondivisorial ideals in F 0 (R). By Lemma 2.2 and Proposition 3.2, ∆ is equal to the union of the set of the canonical ideals and the set G of the I ∈ F 0 (R) such that IT = T . By Lemma 3.1 and Proposition 3.2, the elements of the latter correspond to the subspaces of V /T containing e 0 but not e 3 : hence, we can write G = G 1 ∪ G 2 ∪ G 3 , where G i contains the ideals of G corresponding to subspaces of dimension i.
If I is ⋆-closed for some I ∈ G 3 , then every element of G 3 must be closed, since any other I ′ ∈ G 3 is in the form γI for some γ of valuation 0 (by the first part of the proof); furthermore, every element of G 2 is the intersection of the elements of G 3 containing it, and thus it is ⋆-closed. It follows that ∆(⋆) = ∆ \ {J}; in particular, there is only one such star operation.
Let ⋆ be any star operation different from the three above. Then, ∆(⋆) must contain T and cannot contain any canonical ideal nor any element of G 3 . Hence, ∆(⋆) must be equal to Λ ∪ {T } for some Λ ⊆ G 2 . Moreover, Λ ∪ {T } is equal to ∆(⋆) for some ⋆ if and only if Λ is the (possibly empty) union of equivalence classes under ∼. It follows that |Star(R)| = 2 x + 3, where x is the number of such equivalence classes.