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
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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
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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.
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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.
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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
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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).
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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.
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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.
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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
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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
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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.
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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).
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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
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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)
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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.
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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
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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 .
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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 .
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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)
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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
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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.
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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 .
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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= ∅.