Franz Halter-Koch's Contributions to Factorization Theory ... · History A New Idea Krull Monoids Weakly Krull Monoids C-Monoids Non-Commutative Settings Conference and Special Session

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Franz Halter-Koch'sContributions to Factorization Theory:

The Story of a New Idea

Alfred Geroldinger

Arithmetic and Ideal Theory of

Rings and Semigroups

Graz, 2014

History A New Idea Krull Monoids Weakly Krull Monoids C-Monoids Non-Commutative Settings

Outline

Historical Preliminaries

A New Idea

Krull Monoids

Weakly Krull Monoids

C-Monoids

Non-Commutative Settings


Rings of Integers in Algebraic Number Fields

Let R be a ring of integers in an algebraic number �eld

with (ideal) class group G = C(R).

19th Century:

• Observation: R is factorial if and only if G is trivial.

• Philosophy: The class group G controls the arithmetic of R .

Problem: Describe and classify phenomena of

non-uniqueness of factorizations.


Carlitz and Sets of Lengths

If H is a multiplicatively written, cancellative semigroup and

a = u1 · . . . · uk ∈ H where u1, . . . , uk are irreducibles (atoms), then

• k is called the length of the factorization and

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

For k ∈ N, let ρk(H) denote the supremum over all ` ∈ N such that

there is an equation u1 · . . . · uk = v1 · . . . · v`, with ui , vj irreducible.Equivalently,

ρk(H) = sup{sup L(a) | a ∈ H, k ∈ L(a)} ∈ N≥k ∪ {∞} .

Carlitz (1960): For a ring of integers R there are equivalent:

• ρk(R) = k for all k ≥ 2.

• |G | ≤ 2.


Blocks and Zero-Sum Sequences: The 1960s and 1970s

Let G be an additive abelian group and G0 ⊂ G a subset.

A sequence S = g1 · . . . · gl of group elements from G0 is called a

zero-sum sequence (or a block) if g1 + . . .+ gl = 0.

The set B(G0) of zero-sum sequences over G0 is a monoid with

concatenation as operation. The Davenport constant D(G0) is themaximal length of a minimal zero-sum sequence over G0.

• K. Rogers (1962): If R is a ring of integers, then D(C(R)

)is

the maximal number of prime ideals occurring in the prime

ideal decomposition of aR for an irreducible element a ∈ R .

• H. Davenport (1965): Midwestern Conference on Group

Theory and Number Theory: Study D(G ).

• W. Narkiewicz (1979): Finite abelian groups and

factorization problems:

Study the monoid B(G ) instead of the ring of integers R !


Studies on ρk(R): The 1980s

For small values of k and small class groups, a study of ρk(R) wasstarted by

• A. Czogaªa (1981): Arithmetic characterization of algebraic

number �elds with small class number

• L. Salce and P. Zanardo (1982): Arithmetical characterization

of rings of algebraic integers with cyclic ideal class group

• F. Di Franco and F. Pace (1985)

• K. Feng (1985)

continued later by

• S.T. Chapman, W.W. Smith, and others


Outline


A New Idea

Krull Monoids


C-Monoids



Conference and Special Session

on

Factorization in Integral Domains,

in

Iowa City (1996)

organized by

Daniel D. Anderson

F. Halter-Koch:

Finitely Generated Monoids,

Finitely Primary Monoids, and

Factorization Properties of

Integral Domains


Transfer Homomorphisms

De�nition (Transfer Homomorphism)

A monoid homomorphism θ : H → B is called a transfer

homomorphism if it has the following properties:

(T 1) B = θ(H)B× and θ−1(B×) = H×.

(T 2) If a ∈ H, b1, b2 ∈ B and θ(a) = b1b2, then there exista1, a2 ∈ H such that a = a1a2, θ(a1) ' b1 and θ(a2) ' b2.

Lemma (Transfer Lemma)

Let θ : H → B be a transfer homomorphism. Then we have :

• LH(a) = LB(θ(a)

)for all a ∈ H.

Thus transfer homomorphisms preserve sets of lengths.

• In particular, ρk(H) = ρk(B) for all k ≥ 2.


Strategy for studying the arithmetic

In order to study the arithmetic (e.g, the invariants ρk(H))of a domain or monoid H,

Proceed as follows:

• Study the ideal theory of H.

• Construct a transfer homomorphism θ : H → B .

• Study the arithmetic of B (e.g., study ρk(B)).

Then

ρk(H) = ρk(B) for all k ≥ 2 .


Outline


A New Idea

Krull Monoids


C-Monoids



Krull monoids: De�nition

A monoid H is a Krull monoid if one of the equiv. statements hold:

(a) H is completely integrally closed and v -noetherian.

(b) There is a divisor homomorphism ϕ : H → F = F(P)(For all a, b ∈ H: a | b in H ⇐⇒ ϕ(a) |ϕ(b) in F )

(c) There is a divisor theory ϕ : H → F = F(P)

• For (a): Choinard (1981), P.Wauters (1984): noncommutative

• For (c): Borevic-Safarevic, L. Skula (1970), Gundlach (1972)

• If (a) holds, then I∗v (H) is free abelian with basis X(H).

• (a) ⇐⇒ (c): F. Halter-Koch (1990):

Halbgruppen mit Divisorentheorie


Krull monoids: Examples

• Krause (1990): A domain R is Krull i� R• is Krull.

• Halter-Koch (1993): Analogue for rings with zero-divisors.

• Halter-Koch: Regular congruence submonoids of Krull domains

• Frisch, Reinhart: Monadic submonoids of Int(R), R Krull.

• Wiegand, Herbera, Facchini (∼ 2000): Monoids of modules

If C is a nice class of modules, then V(C) = {[M] | M ∈ C} isKrull (operation stems from the direct sum decomposition)

• Monoids B(G0) of zero-sum sequences over G0:

If G0 is a subset of an additive abelian group G ,

then the embedding

B(G0) ↪→ F(G0)

is a divisor homomorphism, and hence B(G0) is Krull.


Transfer hom. from a general Krull monoid to B(GP)

Suppose the embedding H ↪→ F(P) is a divisor theory.

H −−−−→ F(P)

β

y yβ

B(GP) −−−−→ F(GP)

Then β and its restriction β = β | H are transfer homomorphisms

mapping

a = p1 · . . . · pl ∈ F(P) to S = β(a) = [p1] · . . . · [pl ] ∈ F(GP)

such that

• a is irreducible in H if and only if S is irreducible in B(GP).

• LH(a) = LB(GP)(S).

• In particular, ρk(H) = ρk(B(GP)

)=: ρk(GP) for all k ≥ 2.


First Results

Let G be an abelian group.

• If G0 ⊂ G is �nite, then B(G0) is �nitely generated whence

ρk(H) = ρk(G0) <∞ for all k ≥ 2 .

• Carlitz (1960): If |G | ≤ 2, then ρk(G ) = k for all k ≥ 2.

• If G is in�nite, then ρk(G ) =∞ for all k ≥ 2.

• If 2 < |G | <∞, then

• ρ2k(G ) = kD(G ) for all k ≥ 2.

• kD(G ) + 1 ≤ ρ2k+1(G ) ≤ kD(G ) + D(G )/2.

• Problem 38 in S.T. Chapman and S. GlazNon-Noetherian Commutative Ring Theory (2000):

Do we have kD(G ) + 1 = ρ2k+1(G ) for cyclic groups ?


A Result involving Additive Number Theory

Theorem ( Savchev-Chen, 2007)

Let G be a cyclic group of order |G | = n.

Let U ∈ B(G ) be a minimal zero-sum sequence of length

l ≥⌊n2

⌋+ 2. Then there is some g ∈ G with G = 〈g〉 such that,

with 1 = n1 ≤ n2 ≤ . . . ≤ nl ,

S = (n1g)(n2g) · . . . · (nlg) and n1 + . . .+ nl = n .

Theorem (Gao+G., 2009)

Let H be a Krull monoid with cyclic class group G such that each

class contains a prime divisor. Then for every k ∈ N, we have

ρ2k+1(H) = ρ2k+1(G ) = kD(G ) + 1 .


Outline


A New Idea

Krull Monoids


C-Monoids



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: D.D. Anderson, Mott, Zafrullah (1992)

Weakly Krull monoids: F. Halter-Koch (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).

Examples:

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

• In particular: Non-principal orders in algebraic number �elds


Main Result

F. Halter-Koch:

• Elasticity of factorizations in atomic monoids and integral

domains (1995)

• Finitely generated monoids, �nitely primary monoids and

factorization properties of integral domains (1997)

Theorem

Let H be a weakly Krull monoid that the class group G = C(H) is�nite and each class contains a p ∈ X(H) with p 6⊃ f = (H : H).Then the following statements are equivalent :

(a) ρk(H) <∞ for all k ∈ N.(b) The natural map X(H)→ X(H), P 7→ P ∩ H, is bijective.


Transfer Homomorphisms

Consider

H −−−−→ D = F(P)×T ∼= I∗v (H)

β

y β

yB = B(G ,T , ι) −−−−→ F = F(G )×T

where

• T =∏

p∈P∗(Hp)red and P∗ = {p ∈ X(H) | p ⊃ f}.• 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 .

Then

• B is weakly Krull again, and

• β = β |H is a transfer homomorphism.


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 .


Outline


A New Idea

Krull Monoids


C-Monoids



Conference and Special Session

on

Factorization Properties of

Commutative Rings and Monoids,

in

Chapel Hill (2003)

organized by

Scott T. Chapman

F. Halter-Koch:

C-monoids and congruence

monoids in Krull domains


De�nition and Examples

A monoid H is called a C-monoid if it is a submonoid of a factorial

monoid F such that

H ∩ F× = H× and the class semigroup C∗(H,F ) is �nite.

Examples:

1. Let R be a Mori domain such that f = (R : R) 6= {0}.(a) If C(R) and R/f are �nite, then R is a C-monoid.

(b) In special cases the converse holds (Reinhart)

(c) There is an analogue to (a) in Mori rings with zero-divisors

2. Many congruence monoids, in particular arithmetical

congruence monoids as studied by Chapman, Baginski et al.


C-Monoids: Comments and Strategy

Let H be a Mori monoid such that f = (H : H) 6= {0}.• H is Krull, I∗v (H) is free abelian with (in�nite) basis

v -spec(H) = X(H), and we have a map ϕ : H → I∗v (H).

• Thus there is a factorial monoid F = F××F(P)with (in�nite) basis P such that H ⊂ F .

• The �niteness of the class semigroup C∗(H,F ) allows toconstruct a transfer homomorphism θ : H → B

H −−−−→ F = F××F(P)

θ

y yB −−−−→ F0 = F×

0×F(P0)

where B is a C-monoid again and P0 is �nite.


Main Arithmetical Result for C-Monoids

Studying the C-monoid B and using that ρk(B) = ρk(H) we obtain

Theorem

Let H be a C-monoid.

Then the following statements are equivalent :

(a) ρk(H) <∞ for all k ∈ N.(b) H is simple

(i.e., each minimal H-essential subset of P is a singleton).


Outline


A New Idea

Krull Monoids


C-Monoids



Weak Transfer Homomorphisms

D. Smertnig (2013): Sets of lengths in maximal orders in

central simple algebras

N. Baeth et al. (2014): Factorization of upper triangular matrices

N. Baeth and D. Smertnig: Factorization Theory in


De�nition

A monoid homomorphism θ : H → B is called a weak transfer

homomorphism if it has the following properties:

(T 1) B = B×θ(H)B× and θ−1(B×) = H×.

(WT2) If a ∈ H and b1, . . . , bn are atoms in B such thatθ(a) = b1 · . . . · bn, then there exist atoms a1, . . . , an ∈ H anda permutation σ ∈ Sn such that a = a1 · . . . · an andθ(ai ) = bσ(i) for each i ∈ [1, n].


Transfer and Weak Transfer Homomorphisms

Facts:

• Each transfer homomorphism is a weak transfer

homomorphism but not conversely.

• Weak transfer homomorphisms respect sets of lengths.

Theorem (N. Baeth et al., 2014)

Let R be a commutative BF-domain and Tn(R) the semigroup of

upper triangular matrices with nonzero determinant.

1. det : Tn(R)→ R• is a transfer homomorphism if and only if

R is a principal ideal domain.

2. There is a weak transfer hom. θ : Tn(R)→ R• × . . .× R•.

3. Under a weak assumption on R there is no transfer

homomorphism from T2(R) to any commutative semigroup.


Classical Maximal Orders: D. Smertnig (2013)

Let

• K be a global �eld,

• A a central simple

K -algebra,

• O a holomorphy ring of K ,

• and R a classical maximal

O-order in A

(R subring of A, Z (R) = O,f.g. as O-module, maximal).

A ∼= Mn(D)

. . .R,R ′ . . .

K

OIf every stably free left R-ideal is free, then there exists a transfer

homomorphism

θ : R• → B(CA(O)),

with CA(O) a ray class group of O.


Summary

Based on a solid understanding of the respective ideal theories

transfer homomorphisms

provide a strategy for studying the arithmetic of

• Dedekind domains including rings of integers in number �elds

• Krull domains including integrally closed noetherian domains

• Weakly Krull domains including 1-dimensional noetherian

domains

• C-domains including higher-dimensional noetherian domains R

where R/f and C(R) are �nite

• Maximal orders in central simple algebras over global �elds

(these are non-commutative Dedekind prime rings)

• much much more .... going to come ....

Documents

Franz Halter-Koch's Contributions to Factorization Theory ... · History A New Idea Krull Monoids Weakly Krull Monoids C-Monoids Non-Commutative Settings Conference and Special Session