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= ∅.
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