arithmetic of seminormal weakly krull monoids and...

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Arithmetic of

seminormal weakly Krull monoids and domains

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

International Meeting on Numerical Semigroups

Cortona, September 2014

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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 Weakly Krull Arithmetic Seminormality Main Results Methods

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 · . . . · uk where u1, . . . , uk are irreducibles (atoms),

then k is called the length of the factorization.

• LH(a) = {k | a has a factorization of length k} ⊂ Nis the set of lengths of a.

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

∆(L(a)

)= {k2 − k1, k3 − k2, . . .}

is the set of distances of L(a).

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

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

Sets of distances and unions of sets of lengths

We call

∆(H) =⋃a∈H

∆(L(a)

)⊂ N

the set of distances of H. For k ∈ N, we call

Uk(H) =⋃

k∈L(a)

L(a)

= {` ∈ N | there is an equation u1 · . . . · uk = v1 · . . . · v`}

the union of sets of lengths containing k .

An atomic monoid H is called half-factorial if one foll. equiv. holds:

(a) |L(a)| = 1 for each a ∈ H.

(b) ∆(H) = ∅.(c) Uk(H) = {k} for each k ∈ N.

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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 Weakly Krull Arithmetic Seminormality Main Results Methods

De�nition of Krull monoids

H is called a Krull monoid if 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 :• ϕ is 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.

The divisor class group G is isomorphic to the v -class group:

G = q(F )/q(ϕ(H)

)= {aq

(ϕ(H)

)= [a] | a ∈ F} ∼= Cv (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).

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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 thefollowing equivalent statements are satis�ed :

(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.

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

Finitely primary monoids and domains

A monoid H is called �nitely 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, . . . , ps s.t.

H \ H× ⊂ p1 · . . . · psF and (p1 · . . . · ps)αF ⊂ H .

(b) H is primary, (H : H) 6= ∅ and Hred∼= (Ns

0,+).

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 : R) 6= {0}, then R• is �nitely primary.

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

Weakly Krull monoids and domains

A monoid H is weakly 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 = H.

• We suppose that all weakly Krull monoids are• v -noetherian• Hp are �nitely primary for each p ∈ X(H).

• (H : H) = f 6= ∅.• Example: 1-dim. noeth. domains R s.t. R is a f.g. R-module

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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 Weakly Krull Arithmetic Seminormality Main Results Methods

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],maxU2k(H) = kn, and maxU2k+1(H) = kn + 1.

3. If G is in�nite, then ∆(H) = Uk(H) = N≥2 for all k ∈ N.

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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:• Uk(R) is �nite.

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

• There is no information• on the structure of the set of distances ∆(R)• nor on the structure of the unions Uk(R).

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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 Weakly Krull Arithmetic Seminormality Main Results Methods

Seminormality: De�nitions and Remarks

The seminormalization H ′ of H is de�ned by

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

Then

• H ⊂ H ′ ⊂ H ⊂ q(H).

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

A domain R is seminormal 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)

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

Seminormal �nitely primary monoids

Let H ⊂ H = F = F××[p1, . . . , ps ] be �nitely primary.

• H ′ = p1 · . . . · psF ∪ H ′×.• If F× = {1}, then H ′ ∼= (Ns ∪ {0},+) ⊂ (Ns

0,+).

• A(H ′) ={εpk1

1· . . . · pkss | ε ∈ F×,min{k1, . . . , ks} = 1

}.

• H ′ 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.

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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 Weakly Krull Arithmetic Seminormality Main Results Methods

Algebraic Structure of seminormal weakly Krull monoids

Let H be a seminormal weakly Krull monoid with nontrivial

conductor f = (H : H) ( H, and let P∗ = {p ∈ X(H) | p ⊃ f}.Then we have

1. H is Krull and P∗ is �nite.

2. The monoid I∗v (H) of v -invertible v -ideals satis�es

I∗v (H) ∼= F(P)×∏p∈P∗

(Hp)red ,

and it is seminormal, v -noetherian, and weakly factorial,

3. There is an exact sequence

1→ H×/H× →∐

p∈X(H)

H×p /H×p

ε→ Cv (H)ϑ→ Cv (H)→ 0 .

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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(H)→ X(H) is bijective.

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

1.2 Suppose that ϑ : Cv (H)→ Cv (H) 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(H)→ X(H) is not bijective.

Then for all k ≥ 3, we have

N≥3 ⊂ Uk(H) ⊂ N≥2 .

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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(H)→ X(H) is bijective, and the

homomorphism ϑ : Cv (H)→ Cv (H) is an isomorphism.

where

π : X(H)→ X(H), is de�ned by π(P) = P ∩ H for all P ∈ X(H)

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

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 Weakly Krull Arithmetic Seminormality Main Results Methods

Transfer Homomorphisms

Consider

H −−−−→ D = F(P)×T ∼= I∗v (H)

β

y β

yB = B(G ,T , ι) −−−−→ F = F(G )×T

where

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

• ι : T → G is de�ned by ι(t) = [t].

• β : D → F be the unique homomorphism satisfying β(p) = [p]for all p ∈ P and β |T = idT .

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

2. Transfer homomorphisms preserve sets of lengths. In

particular, unions of sets of lengths and half-factoriality.

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

Combinatorial weakly Krull monoids: B(G ,T , ι)Let G be a �nite abelian group and T = D1 × . . .× Dn 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 de�ned by ι.Special Cases:

• If G = {0}, then B(G ,T , ι) = T = D1 × . . .× Dn

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 .

Sets of Lengths Weakly Krull Arithmetic Seminormality Main Results Methods

Saturated submonoids inherit

the properties under consideration

Consider a saturated submonoid

H ⊂ D = F(P)×n∏

i=1

Di ,

where P ⊂ D is a set of primes, n ∈ N0, and

D1, . . . ,Dn 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 : H) 6= ∅.