Arithmetic of seminormal weakly Krull monoids and domains

seminormal weakly Krull monoids and domains

A. Geroldinger^∗ and F. Kainrath and A. Reinhart

International Meeting on Numerical Semigroups

Cortona, September 2014

Outline

(Unions of) Sets of Lengths Krull and weakly Krull monoids

Arithmetic: Precise versus Qualitative Results Seminormal Monoids and Domains

Main Results

Methods: Transfer Homomorphisms

Sets of lengths in monoids

Let H be a multiplicatively written, commutative, cancellative semigroup, and let a ∈ H be a non-unit.

• If a = u₁· . . . ·u_k where u₁, . . . ,u_k are irreducibles (atoms), then k is called thelengthof the factorization.

• L_H(a) = {k | a has a factorization of length k} ⊂ N is the set of lengthsof a.

• If L(a) = {k₁,k₂,k₃, . . .}with k₁<k₂ <k₃ < . . ., then

∆ L(a)
= {k₂−k₁,k₃−k₂, . . .}

is the set of distancesof L(a).

• If |L(a)| ≥ 2, then |L(a^m)| >m for each m ∈ N.

Sets of distances and unions of sets of lengths

We call

∆(H) = [

a∈H

∆ L(a)
⊂ N theset of distances of H. For k ∈ N, we call

U_k(H) = [

k∈L(a)

L(a)

= {` ∈ N | there is an equation u1· . . . ·u<sub>k</sub> =v1· . . . ·v`} theunion of sets of lengths containing k.

An atomic monoid H is calledhalf-factorialif one foll. equiv. holds:

(a) |L(a)| = 1 for each a ∈ H.

(b) ∆(H) = ∅.

(c) U_k(H) = {k} for each k ∈ N.

Outline

(Unions of) Sets of Lengths Krull and weakly Krull monoids

Arithmetic: Precise versus Qualitative Results Seminormal Monoids and Domains

Main Results

Methods: Transfer Homomorphisms

Denition of Krull monoids

H is called aKrull monoidif one of the foll. equiv. holds : (a) H is v-noetherian and completely integrally closed.

(b) H has a divisor theory ϕ: H → F(P) = F :

• ϕis a divisor homomorphism:

For all a, b ∈ H we have a | b if and only if ϕ(a) | ϕ(b) .

• For all p ∈ P there is a set X ⊂ H such that p = gcd(ϕ(X ))).

(c) There is a divisor homomorphism into any free abelian monoid.

Thedivisor class groupG is isomorphic to thev-class group:

G = q(F )/q ϕ(H)
= {aq ϕ(H)
= [a] | a ∈ F } ∼= C_v(H) . Let R be a domain.

• R is a Krull domain i ^• is a Krull monoid.

• Integrally closed noetherian domains are Krull by Property (a).

Primary monoids and domains

1. An element q ∈ H is called primary if q /∈ H^× and, for all a, b ∈ H, if q | ab and q - a, then q | bⁿ for some n ∈ N.

2. H is called primary if m = H \ H^×6= ∅ and one of the following equivalent statements are satised :

(a) s-spec(H) = {∅, H \ H^×}. (b) Every q ∈ m is primary.

(c) For all a, b ∈ m there exists some n ∈ N such that a | bⁿ. 3. Let R be a domain.

Then R^• is primary i R is one-dimensional and local.

Finitely primary monoids and domains

A monoid H is callednitely primary (of rank s and exponent α) if one of the following equivalent conditions holds:

(a) There exist s, α ∈ N with the following properties :

H is a submonoid of a factorial monoid F = F^××[p1, . . . ,ps] with s pairwise non-associated prime elements p₁, . . . ,p_s s.t.

H \ H^×⊂p₁· . . . ·p_sF and (p₁· . . . ·p_s)^αF ⊂ H . (b) H is primary, (H : bH) 6= ∅ and bH_red∼= (N^s₀, +).

Clearly, numerical monoids are nitely primary of rank 1.

Let R be a domain.

• If R is a one-dimensional local Mori domain such that (R : bR) 6= {0}, then R^• is nitely primary.

Weakly Krull monoids and domains

A monoid H isweakly Krull if

H = \

p∈X(H)

Hp and {p ∈ X(H) | a ∈ p} is nite for all a ∈ H ,

Weakly Krull domains: Anderson, Mott, and Zafrullah, 1992 Weakly Krull monoids: Halter-Koch, Boll. UMI 1995

• A domain R is weakly Krull i R^• is a weakly Krull monoid.

• H v-noetherian: H weakly Krull ⇐⇒ v-max(H) = X(H).

• H Krull ⇒ H seminormal v-noetherian weakly Krull a. H = bH.

• We suppose that all weakly Krull monoids are

• v-noetherian

• Hp are nitely primary for each p ∈ X(H).

• (H : bH) = f 6= ∅.

• Example: 1-dim. noeth. domains R s.t. R is a f.g. R-module

Outline

(Unions of) Sets of Lengths Krull and weakly Krull monoids

Arithmetic: Precise versus Qualitative Results Seminormal Monoids and Domains

Main Results

Methods: Transfer Homomorphisms

Arithmetic of Krull monoids: Precise Results

Let H be a Krull monoid with class group G such that each class contains a prime divisor.

1. (Carlitz 1960) H is half-factorial if and only if |G| ≤ 2.

2. Let 2 < |G| < ∞. Then

• ∆(H) is a nite interval with min ∆(H) = 1.

• All Uk(H) are nite intervals.

• .... and much more .... for example ....

• If G is cyclic of order n, then ∆(H) = [1, n − 2], max U2k(H) = kn, and max U2k+1(H) = kn + 1.

3. If G is innite, then ∆(H) = U_k(H) = N≥2 for all k ∈ N.

Arithmetic of weakly Krull monoids: Qualitative Results

Let R be a non-principal order in an algebraic number elds with Picard group G.

• Apart from quadratic number elds (Halter-Koch 1983), there is no characterization of half-factoriality.

• ∆(R) is nite. If |G| ≤ 2, then it is open whether 1 ∈ ∆(R).

• For each k ∈ N≥2 the following are equivalent:

• U_k(R) is nite.

• The natural map X(bR) → X(R) is bijective.

• There is no information

• on the structure of the set of distances ∆(R)

• nor on the structure of the unions U_k(R).

Outline

(Unions of) Sets of Lengths Krull and weakly Krull monoids

Arithmetic: Precise versus Qualitative Results Seminormal Monoids and Domains

Main Results

Methods: Transfer Homomorphisms

Seminormality: Denitions and Remarks

TheseminormalizationH⁰ of H is dened by

H⁰ = {x ∈ q(H) | there is some N ∈ N such that xⁿ∈H for all n ≥ N}

Then

• H ⊂ H⁰⊂ bH ⊂ q(H).

• H is seminormal if H = H⁰. Equivalently, if x ∈ q(H) and x²,x³ ∈H, then x ∈ H.

A domain R isseminormal if one of the foll. equiv. holds:

(a) R^• is seminormal.

(b) Pic(R) → Pic R[X ]
is an isomorphism.

Traverso (1970), Swan (1980); Survey by Vitulli (2010)

Seminormal nitely primary monoids

Let H ⊂ bH = F = F^××[p₁, . . . ,p_s]be nitely primary.

• H⁰ =p1· . . . ·psF ∪ H^0×.

• If F^×= {1}, then H⁰ ∼= (N^s∪ {0}, +) ⊂ (N^s₀, +).

• A(H⁰) =εp₁^k1 · . . . ·p_s^ks | ε ∈F^×,min{k₁, . . . ,k_s} =1 .

• H⁰ is seminormal, v-noetherian, and

nitely primary of rank s and exponent 1.

For a domain R the following statements are equivalent : (a) R is a seminormal one-dimensional local Mori domain.

(b) R^• is seminormal nitely primary.

Outline

(Unions of) Sets of Lengths Krull and weakly Krull monoids

Arithmetic: Precise versus Qualitative Results Seminormal Monoids and Domains

Main Results

Methods: Transfer Homomorphisms

Algebraic Structure of seminormal weakly Krull monoids

Let H be a seminormal weakly Krull monoid with nontrivial conductor f = (H : bH) ( H, and let P^∗= {p ∈ X(H) | p ⊃ f}.

Then we have

1. H is Krull and Pb ^∗ is nite.

2. The monoid I_v^∗(H) of v-invertible v-ideals satises I_v^∗(H) ∼= F (P) × Y

p∈P^∗

(Hp)_red,

and it is seminormal, v-noetherian, and weakly factorial, 3. There is an exact sequence

1 → bH^×/H^×→ a

p∈X(H)

Hbp^×/Hp^×

→ Cε _v(H)→ C^ϑ _v( bH) → 0 .

Arithmetic Structure

Suppose in addition that G = Cv(H) is nite, and that every class contains a p ∈ X(H) with p 6⊃ f.

1. Suppose the natural map X(bH) → X(H) is bijective.

1.1 U_k(H) is a nite interval for all k ≥ 2.

1.2 Suppose that ϑ: Cv(H) → Cv( bH) is an isomorphism.

Then there is a transfer homomorphism θ : H → B(G).

In particular, (unions of) sets of lengths and (monotone) catenary degrees of H and B(G) coincide.

2. Suppose the natural map X(bH) → X(H) is not bijective.

Then for all k ≥ 3, we have

N≥3 ⊂ U_k(H) ⊂ N≥2.

Characterization of Half-Factoriality

Suppose in addition that the class group G = C(H) is nite, and that every class contains a p ∈ X(H) with p 6⊃ f.

Then the following statements are equivalent : (a) c(H) ≤ 2.

(b) H is half-factorial.

(c) |G| ≤ 2, the natural map X(bH) → X(H) is bijective, and the homomorphism ϑ: C_v(H) → C_v( bH) is an isomorphism.

where

π : X( bH) → X(H), is dened by π(P) = P ∩ H for all P ∈ X(bH)

Outline

(Unions of) Sets of Lengths Krull and weakly Krull monoids

Arithmetic: Precise versus Qualitative Results Seminormal Monoids and Domains

Main Results

Methods: Transfer Homomorphisms

Transfer Homomorphisms

Consider

H −−−−→ D = F(P)×T ∼= I_v^∗(H)

β





y ^βe



 y B = B(G, T , ι) −−−−→ F = F(G)×T where

• H ,→ D is saturated, and the class group G = C(H, D) satises G = {[p] | p ∈ P} ⊂ G.

• ι : T → G is dened by ι(t) = [t].

• β :e D → F be the unique homomorphism satisfying eβ(p) = [p]

for all p ∈ P and eβ | T = id_T.

1. The restriction β = eβ | H : H → B is a transfer hom.

2. Transfer homomorphisms preserve sets of lengths. In particular, unions of sets of lengths and half-factoriality.

Combinatorial weakly Krull monoids: B(G, T , ι)

Let G be a nite abelian group and T = D₁× . . . ×D_n a monoid.

Let

• ι : T → G a homomorphism, and

• σ : F (G) → G satisfying σ(g) = g.

Then

B(G, T , ι) = {S t ∈ F(G)×T | σ(S) + ι(t) = 0 } ⊂ F(G)×T the T -block monoid over G dened by ι.

Special Cases:

• If G = {0}, then B(G, T , ι) = T = D₁× . . . ×D_n is a nite product of nitely primary monoids.

• If T = {1}, then

B(G, T , ι) = B(G) = {S ∈ F(G) | σ(S) = 0} ⊂ F(G) is the monoid of zero-sum sequences over G.

Saturated submonoids inherit the properties under consideration

Consider a saturated submonoid

H ⊂ D = F(P)×Yⁿ

i=1

D_i,

where P ⊂ D is a set of primes, n ∈ N₀, and D₁, . . . ,D_n are primary monoids. Then we have .

1. If C(H, D) is a torsion group, then H is a weakly Krull monoid.

2. If D1, ...,Dn are seminormal nitely primary, then H is seminormal and v-noetherian with (H : bH) 6= ∅.