Arithmetic of Krull monoids ()Ideals = F (spec )); Dedekind implies ’: R !Ideals : a 7!aR is a divisor homomorphism. De nitions Problems Zero-Sums Sets of Lengths Quantitative

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Arithmetic of Krull monoids

Alfred Geroldinger

Institute of Mathematics and Scienti�c Computing

University of Graz

Additive Combinatoricsin Paris, 2012

De�nitions Problems Zero-Sums Sets of Lengths Quantitative Aspects Beyond I Beyond II

Outline

De�nitions and Examples

Problems

Monoids of Zero-sum sequences

Sets of Lengths

Quantitative Aspects

Beyond Sets of Lengths

Beyond Krull monoids


Zero-Sum Sequences form a semigroup

Let G = (G ,+) be an additively written abelian group.

• A sequence S = (g1, . . . , g`) over G : �nite, unorderedsequence of terms from G , repetition allowed.

• S has sum zero if σ(S) = g1 + . . .+ g` = 0.

• The set of sequences forms a semigroup with concatenation ofsequences as the operation: (F(G ), ·). Indeed

if T = (h1, . . . , hk), then ST = (h1, . . . , hk , g1, . . . , g`) .

• The set of zero-sum sequencesB(G ) = {S ∈ F(G ) | σ(S) = 0} is a subsemigroup.

• Unit element in B(G ): empty sequence.


• Divisibility in F(G ): T | S ⇐⇒ T is a subsequence of S⇐⇒ there exists an T ′ such that S = TT ′.

• If S and T have sum zero, then T ′ has sum zero: in otherwords,

T | S in B(G ) if and only if T | S in F(G ) .

PROBLEM: Factorize a zero-sum sequence S into minimalzero-sum sequences, say

S = S1 · . . . · Sk ,

and study the set L(S) of all possible k .

WHY ?


De�nition of Krull monoids

A monoid H is a Krull monoid if one of the following equivalentconditions is satis�ed:

• There is a homomorphism ϕ : H → F(P) such that, for alla, b ∈ H, we have

a | b in H if and only if ϕ(a) |ϕ(b) in F(P) .

(ϕ is called a divisor homomorphism).

• H is completely integrally closed and satis�es theascending chain condition on divisorial ideals.

Then

• N = F(P); Every monoid F(P) (factorial monoid) is Krull.

• B(G ) ↪→ F(G ) is Krull.

• R Dedekind ⇐⇒ Ideals = F(spec•(R)); R Dedekind impliesϕ : R• → Ideals : a 7→ aR is a divisor homomorphism.


More Examples: Ring theory

Domains: Let R be a domain. Then (R•, ·) is a monoid.

• R is a Krull domain if and only if R• is a Krull monoid.

• The monoid algebra R[H] is Krull if and only if R is Krull andH is Krull (for a reduced monoid H; Chouinard 1981)).

• Integrally closed noetherian domains are Krull.• One-dimensional Krull: Dedekind domains• Higher dimensional Krull: a�ne K -algebras, rings of invariants

Submonoids of Domains: Regular congruence monoids in Krulldomains are Krull.Example: Let R be a non-principal order in a Dedekind domain R

with conductor f = (R :R). Then

H = {a ∈ R• | aR + f = R}

is a Krull monoid.


More Examples: Module Theory

Let R be a commutative ring, C a class of R-modules closed underisomorphisms, �nite direct sums and direct summands.For a module M, let [M] denote its isomorphism class.Then

H = {[M] | M ∈ C}is an additive semigroup where addition is de�ned as

[M] + [N] = [M ⊕ N] .

If C is the class of noetherian R-modules, then every M ∈ C is a�nite direct sum of indecomposable R-modules.

Theorem

• (Krull-Schmidt 1930s) If EndR(M) is local for all M ∈ C(e.g., all M have �nite length), then H is factorial.

• (Wiegand, Facchini, 2000s) If EndR(M) is semilocal for all

M ∈ C, then H is a Krull monoid.


Class groups I

There are equivalent:

• H is a Krull monoid.

• H has a divisor theory. This is a divisor homomorphismϕ : H → D = F(P) such that, for every p ∈ P , there is a �nitesubset X ⊂ H such that p = gcd

(ϕ(X )

).

Divisor theories are unique (up to isomorphism).If ϕ = (H ↪→ D), then

G = q(D)/q(H) = {aq(H) = [a] | a ∈ D}

is called the class group of G , and

GP = {[p] | p ∈ P} ⊂ G

denotes the set of classes containing prime divisors.


Class Groups II

FACT 1: A reduced Krull monoid is uniquely determined by itsclass group and by the number of prime divisors in the classes.

Note: If R is a ring of integers in an algebraic number �eld. Thenevery class contains in�nitely many prime ideals. Thus Fact 1justi�es the

Classical Philosophy: The class group determines the arithmetic.

FACT 2: Let G = (G ,+) be an abelian group with |G | 6= 2. ThenB(G ) is the unique Krull monoid with class group (isomorphic to )G where every class contains precisely one prime divisor.


Outline


Problems


Sets of Lengths





Sets of lengths: First observations

Let H be a Krull monoid. Every nonunit a ∈ H can be written as

a = u1 · . . . · uk where u1, . . . , uk ∈ A(H) .

• A(H) is the set of atoms (irreducible elements).

• k is called the length of the factorization.

•LH(a) = {k | a has a factorization of length k} ⊂ N

is the set of lengths of a.

• All sets of lengths are �nite: if H ⊂ F(P) anda = p1 · . . . · p` ∈ H, then max L(a) ≤ `.

• L(H) = {L(a) | a ∈ H} is the system of all sets of lengths.


Sets of lengths: Classical results

Let H be a Krull monoid with class group G , and suppose thatevery class has a prime divisor. We have

• (19th Century) H is factorial i� |G | = 1.

• (Carlitz 1960) |G | ≤ 2 if and only if |L(a)| = 1 for all a ∈ H.

• (�liwa 1982) If |G | ≥ 3, then for every m ∈ N, there is anam ∈ H with |L(am)| = m.

Note: If a = u1 · . . . · uk = v1 · . . . · v` with ui , vj ∈ A(H), then

am = (u1 · . . . · uk)i (v1 · . . . · v`)m−i for all i ∈ [0,m]

and henceL(am) ⊃ {`m + i(k − `) | i ∈ [0,m]} .


Narkiewicz 1979: There is a Transfer Homomorphism

from a general Krull monoid to B(G )

Let H be a Krull monoid, ϕ = (H ↪→ F(P)) its divisor theory, G itsclass group, and GP = G . Consider

H −−−−→ F(P)

β

y yβ̃

B(G ) −−−−→ F(G )

Then β = β̃ | H is a transfer hom., where β̃ maps an element

a = p1 · . . . · pl ∈ F(P) to β̃(a) = [p1] · . . . · [pl ] ∈ F(G ) ,

and we have

• a ∈ H ⇐⇒ β̃(a) is a zero-sum sequence.

• LH(a) = LB(G)(β(a))


Main Question

Let H be a Krull monoid with class group G such that every classcontains a prime divisor.

FACT : Sets of lengths in H can be studied in theassociated monoid of zero-sum sequences B(G ).

HOW DO SETS OF LENGTHS LOOK LIKE ?

How do we get started ?We have

UV = W1 · . . . ·Wm

(g1, . . . , gk)(h1, . . . , hl ) = (g1, g5, h3)(g2, h4, h7, h7) . . . . . . ,

where U,V ,Wj are atoms in B(G ).In other words, they are minimal zero-sum sequences over G .


Outline


Problems


Sets of Lengths





Let G0 ⊂ (G ,+) and S = g1 · . . . · gl ∈ F(G0) a sequence over G0.

• |S | = l is the length of S ,

• σ(S) = g1 + . . .+ gl ∈ G is the sum of S ,

• Σk(S) = {∑

i∈I gi | ∅ 6= I ⊂ [1, l ] with |I | = k} ⊂ G isthe set of k-term subsums of S ,

• Σ(S) =∑

k≥1 Σk(S) is the set of (all) subsums of S .

Furthermore,

• S is called a zero-sum sequence of σ(S) = 0,

• S is called zero-sum free if 0 /∈ Σ(S),

• B(G0) ↪→ F(G0) monoid of zero-sum sequences over G0,

• A(G0) := A(B(G0)

)the set of atoms over G0,

in other words, the minimal zero-sum sequences over G0,

• D(G0) = max{|U| | U ∈ A(G0)} ∈ N isDavenport constant of G0. Equivalently, we have

• D(G ) is the smallest integer ` such that every sequence oflength at least ` has a nontrivial zero-sum subsequence.


Methods: Group algebras

Let R be a domain. Let d(G ,R) denote the largest integer l ∈ Nhaving the following property:

There is some sequence S = g1 · . . . · g` such that

(a1−X g1)·. . .·(al−X g`) 6= 0 ∈ R[G ] for all a1, . . . , al ∈ R• .

If S is zero-sum free, then all these expressions are nonzero(the coe�cient of X 0 is nonzero), and hence

D(G )− 1 ≤ d(G ,R) .

Note: Let G ,G ′ be two �nite abelian groups.

• If K is a splitting �eld, then K [G ] ∼= K |G | ∼= K [G ′], but wemay have D(G ) 6= D(G ′). HOWEVER,

• (Higman) Z[G ] ∼= Z[G ′] implies that G ∼= G ′.


Methods: Alon's Combinatorial Nullstellensatz 1999

Let R be a domain, A1, . . . ,An ⊂ R �nite nonempty subsets,f ∈ R[X ], and gi =

∏a∈Ai

(Xi − a) ∈ R[Xi ] for all i ∈ [1, n].Then the following statements are are equivalent:

• f (a1, . . . , an) = 0 for all (a1, . . . , an) ∈ A1 × . . .× An.

• There are h1, . . . , hn ∈ R[X ] with deg(gi ) + deg(hi ) ≤ deg(f )such that

f =n∑

i=1

gihi .

Many applications; many variations and extensions (Ball-Serra,Kouba, Heinig, Michalek,..)


Methods: Addition Theorems

Theorem

Let A,B ⊂ G be �nite nonempty subsets, H = Stab(A + B) be the

stabilizer of the sumset A + B, and ΦH : G → G/H.

(a) (Kneser) |ΦH(A) + ΦH(B)| ≥ |ΦH(A)|+ |ΦH(B)| − 1.

(b) (Kemperman-Scherk)

|A + B| ≥ |A|+ |B| −min{ rA,B(g) | g ∈ A + B} .(c) (Grynkiewicz 2005) Partition Theorem.

(d) (DeVos-Goddyn-Mohar 2009) A generalization of Kneser's ....

Corollary (to (b))

If S = S1S2 is zero-sum free, then |Σ(S)| ≥ |Σ(S1)|+ |Σ(S2)|.


Methods: Inductive Method I

Corollary (to Grynkiewicz or DeVos-Goddyn-Mohar)

Let S ∈ F(G ), n ∈ [1, |S |], and H = Stab(Σn(S)). Then

|Σn(S)| ≥( ∑g∈G/H

min{n, vg(φH(S)

)} − n + 1

)|H| .

To study a given sequence S = g1 · . . . · g` proceed as follows:

• Find a suitable subgroup K ⊂ G , consider the naturalepimorphism ϕ : G → G/K , and ϕ(S) = ϕ(g1) · . . . · ϕ(g`).

• Consider a factorization S = S0S1 · . . . · Sk such that |Si | issmall and ϕ(Si ) ∈ B(G/K ) for all i ∈ [1, k].

• Investigate the sequences T = σ(S1) · . . . · σ(Sk) ∈ F(K ) andS0T ∈ F(G ). Clearly, if S is zero-sum free, then S0T iszero-sum free too.


Methods: Inductive Method II

Let η(G ) resp. s(G ) denote the smallest integer l ∈ N with thefollowing property:

• Every sequence S over G of length |S | ≥ l has azero-sum subsequence T of length |T | ∈ [1, exp(G )] resp.zero-sum subsequence T of length |T | = exp(G )

For the Erd®s-Ginzburg-Ziv constant s(G ) we have

s(G ) ≥ η(G ) + exp(G )− 1 . (∗)

Theorem

• (Gao) η(G ) ≤ |G | and s(G ) ≤ |G |+ exp(G )− 1.If exp(G ) ≤ 4 or exp(G ) is large, then equality in (∗)

• If G = Cn1 ⊕ Cn2 , then s(G ) = η(G ) + n2 − 1 = 2n1 + 2n2 − 3(n1 = 1: EGZ-Theorem; n1 = n2: Reiher).

• Groups of higher rank: remember Bhowmik's talk (yesterday).


The Davenport constant again

Let G = Cn1 ⊕ . . .⊕ Cnrwhere 1 < n1 | . . . | nr . Then

D∗(G ) := 1 +r∑

i=1

(ni − 1) ≤ D(G ) .

• (Olson, Kruyswijk 1960s): Equality for p-groups and rank 2groups.

• Very limited progress for groups of small rank and groups closeto p-groups: Bhowmik, Gao, Schlage-Puchta, Schmid,....

• Conjecture: Equality for rank three groups and for G = C rn .

• (G.+Schneider) For every r ≥ 4 there are in�nitely manygroups G of rank r for which inequality holds.

• (Liebmann-G.-Philipp, 2012) Inequality for G = C2 ⊕ C r2n with

n odd, r ≥ 4.


Structure of minimal zero-sum sequences: Cyclic groups

Theorem ( Savchev-Chen 2007)

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

1. Let S ∈ F(G ) be zero-sum free of length l ≥ n+12 .

Then there are g ∈ G with G = 〈g〉 and 1 = n1 ≤ . . . ≤ n`with m = n1 + . . .+ n` < |G | such that

S = (n1g)(n2g) · . . . · (nlg) and Σ(S) = {g , 2g , . . . ,mg} .

2. 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 .


Structure of minimal zero-sum sequences: Rank two groups

Let G = Cm ⊕ Cmn with m, n ∈ N and m ≥ 2. A sequence S overG of length D(G ) = m + mn − 1 is a minimal zero-sum sequence ifand only if it has one of the following two forms :

•

S = eord(e1)−11

ord(e2)∏ν=1

(xνe1 + e2) , where

{e1, e2} is a basis of G , x1, . . . , xord(e2) ∈ [0, ord(e1)− 1], andx1 + . . .+ xord(e2) ≡ 1 mod ord(e1).

•

S = g sm−11 g(n−s)m+12

m−1∏ν=1

(−xνg1 + g2) , where

{g1, g2} is a generating set of G with ord(g2) = mn, s ∈ [1, n],x1, . . . , xm−1 ∈ [0,m − 1], x1 + . . .+ xm−1 = m − 1, and(s = 1 or mg1 = mg2

).


Proof of the Structural Result

The result was proved in four papers:

• The prime case G = Cp ⊕ Cp: Christian Reiher,to appear in Journal of the London Math. Soc.

• Multiplicity by two: If the Structural Result holds for Cn ⊕ Cn,then it holds for C2n ⊕ C2n.W. Gao and A. G.: Integers 3 (2003)

• Multiplicity by odd numbers:

W. Gao and A.G. and D.J. Grynkiewicz:Acta Arith. 141 (2010), 103 � 152

• General Groups of Rank Two: W.A. Schmid:Acta Arith. 143 (2010), 333 � 343.

Partial results and special cases: Bhowmik, Halupczok,Schlage-Puchta; Fang Chen, Svetoslav Savchev.


Outline


Problems


Sets of Lengths





Set of distances: De�nition

• For a �nite subset L = {a1, . . . , at} ⊂ Z with a1 < . . . < at let

∆(L) = {aν+1 − aν | ν ∈ [1, t − 1]} ⊂ N

denote the set of (successive) distances of L.

•∆(H) =

⋃a∈H

∆(L(a)

)⊂ N

denotes the set of distances of H.

By de�nition we have

• ∆(H) = ∅ ⇐⇒ all sets of lengths are singletons,

• ∆(H) = {d} ⇐⇒ All sets of lengths are arithmeticalprogressions with di�erence d .


Set of distances: Properties

For a subset G0 ⊂ (G ,+) we set

∆(G0) = ∆(B(G0)

).

• For every monoid H, we have min∆(H) = gcd∆(H).

• max∆(G0) ≤ D(G0)− 2.

• (Carlitz 1960) If |G | ≤ 2, then ∆(G ) = ∅.

• If 2 < |G | <∞, then ∆(G ) is an interval with 1 ∈ ∆(G ).

• If G is in�nite, then ∆(G ) = N.


Re�ned elasticities

For a monoid H and k ∈ N, let

ρk(H) = max{` | there exists an equation u1 · . . . · uk = v1 · . . . · v`}

A simple counting argument shows that for ρk(G ) := ρk

(B(G )

)• ρ2k(G ) = kD(G ).

• 1 + kD(G ) ≤ ρ2k+1(G ) ≤ kD(G ) +⌊D(G)2

⌋.

The upper bound can be realized for "many" groups withD∗(G ) = D(G ). However, using Savchev-Chen

Theorem (Gao+G.)

Let G be a cyclic group. Then for every k ∈ N, we have

ρ2k+1(G ) = kD(G ) + 1 .


AAMPs: Almost Arithmetical Multiprogressions

Let d ∈ N, M ∈ N0 and {0, d} ⊂ D ⊂ [0, d ].A subset L ⊂ Z is called an

AAMP with di�erence d , period D, and bound M,if

L = y + (L′ ∪ L∗ ∪ L′′) ⊂ y +D + dZ

where

• y ∈ Z is a shift parameter,

• L′ ⊂ [−M,−1] and L′′ ⊂ max L∗ + [1,M], are the (short)beginning and end parts of L,

• L∗ is (the ) �nite nonempty (important and long middle part)with min L∗ = 0 and

L∗ = (D + dZ) ∩ [0,max L∗] .


Structure Theorem - Realization Theorem

Theorem (Freiman, G., Halter-Koch,...)

Let H be a Krull monoid with �nite class group G.

Then there is an M = M(G ) ∈ N0 such that every set of lengths is

an AAMP with di�erence d ∈ ∆(G ) and bound M.

Theorem (Schmid 2009)

Let M ∈ N0 and ∆ ⊂ N be a �nite nonempty set. Then there

exists a Krull monoid H with �nite class group such that :

For every AAMP L with di�erence d ∈ ∆ and bound M there is

some yH,L ∈ N such that

y + L ∈ L(H) for all y ≥ yH,L.

Indeed, there exists an algebraic number �eld such that its ring of

integers has this property.


An interlude

Theorem (Kainrath)

Suppose that H is a Krull monoid with in�nite class group, and

that every class contains a prime divisor.

Then every �nite subset L ⊂ N≥2 can be realized as a set of

lengths of H.

Theorem (Frisch)

Suppose that H = Int(Z) is the ring of integer valued polynomials.

Then every �nite subset L ⊂ N≥2 can be realized as a set of

lengths of H.


On the parameters M and ∆

The bound M: Only very rough upper bounds are known.

On the set of di�erences ∆ in long AAMPs:By the proof of the Structure Theorem, this set ∆ is equal to

∆∗(G ) = {min∆(G0) | G0 ⊂ G ,∆(G0) 6= ∅} ⊂ ∆(G ) .

There is some work on ∆∗(G ). Recall that |G | ≥ 3.

SIMPLE FACTS:

• 1 ∈ ∆∗(G ).

• If g ∈ G with ord(g) ≥ 3, then ord(g)− 2 ∈ ∆∗(G ).

• If r(G ) ≥ 2, then [1, r(G )− 1] ⊂ ∆∗(G ).


Cross numbers and ∆(G0)

For a sequence S = g1 · . . . · gl let

k(S) =∑̀i=1

1ord(gi )

denote the cross number of S ,

and de�ne (Krause 1984)

K(G ) = max{k(S) | S ∈ A(G )} denote the cross number of G ,

For p-groups, the precise value of K(G ) is known !!There are equivalent: (Skula, Zaks 1970s)

• ∆(G0) = ∅.• k(U) = 1 for all U ∈ A(G0).


On ∆∗(G )

Theorem

• (Schmid) max∆∗(G ) = max{exp(G )− 2,m(G )} and

m(G ) ≤ max{r∗(G )− 1,K(G )− 1} .

If G is a p-group, then K(G ) < r∗(G ), m(G ) = r(G )− 1and thus max∆∗(G ) = max{exp(G )− 2, r(G )− 1}.

• There are results for groups with large rank and for groups

with large exponent.

• (G. + Hamidoune 2002)

max(∆∗(Cn) \ {n − 2}

)=⌊n2

⌋− 1 .

• (Plagne + Schmid 2012) Detailed study of ∆∗(Cn).


Arithmetical Characterizations of Class Groups

Let R be a ring of integers with class group G .

Classical Philosophy: The class group determines the arithmetic.

• R is factorial i� |G | = 1.

• Carlitz 1960: |L(a)| = 1 for all non-zero a ∈ R i� |G | ≤ 2.

Narkiewicz 1970s:

• What about the converse:Do arithmetical phenomena characterize the class group ?

• Give arithmetical characterizations of the class group.

Positive answers by Kaczorowski, Halter-Koch, Rush, and others.Let

L(G ) := L(B(G )

)= {L(A) | A is a zero-sum sequence over G}

denote the system of sets of lengths over G .


Systems of Sets of Lengths

PROBLEM: Given two �nite abelian groups G ,G ′ such thatL(G ) = L(G ′). Does it follow that G ∼= G ′?

Apart from

L(C1) = L(C2) and L(C3) = L(C2 ⊕ C2) ,

we get, by using ALL what we had so far,

Theorem

Let G and G ′ be �nite abelian groups with D(G ) ≥ 4, and suppose

that L(G ) = L(G ′).

1. If G is cyclic or an elementary 2-group, then G ∼= G ′.

2. (Schmid) If G = Cn ⊕ Cn with n ≥ 3, then G ∼= G ′.

3. (Baginski-G.-Grynkiewicz-Philipp) If G has rank two and

D(G ′) = D∗(G ′), then G ∼= G ′.


Outline


Problems


Sets of Lengths





Quantitative aspects of non-unique factorizations

Let R be the ring of integers in an algebraic number �eld, G itsideal class group and |G | ≥ 3.

Narkiewicz 1960s: Study the asymptotic behaviour of the followingcounting functions:

Fk(x) = #{aR | a ∈ R•, (R :aR) ≤ x and |Z(a)| ≤ k}Gk(x) = #{aR | a ∈ R•, (R :aR) ≤ x and |L(a)| ≤ k}Mk(x) = #{aR | a ∈ R•, (R :aR) ≤ x and max L(a) ≤ k}

Using the Tauberian Theorem of Ikehara and Delange, one canshow that that all these quantities behave for x →∞asymptotically like

x(log x)−A(log log x)B

with exponents A, B ∈ R>0 depending only on G .


FURTHER DIRECTIONS:

1. Better asymptotics: Recent progress by Kaczorowski, Perelli,Radziejewski.

2. Determine the exponents A,B .

3. Non-principal orders.

In particular, we have

Fk(x) ∼ x(log x)−1+1/|G |(log log x)Nk(G) , where

Nk(G ) is the maximal length of an ordered zero-sum sequencehaving at most k distinct factorizations. We have

•∑r

i=1 ni ≤ N1(G ) ≤ N2(G ) ≤ .....• Conjecture(Narkiwiecz-Sliwa 1982):

∑ri=1 ni = N1(G )


Theorem (Gao et al., 2011,2012,201?)

(a) If k is small with respect to n, then Nk(Cn) = n

(use Savchev-Chen).

(b) N1(Cn ⊕ Cn) = 2n.

(c) If D(C 3n1

) ≤ 3n1 − 1, then N1(Cn1 ⊕ Cn2) = n1 + n2.

Method for (b):

• Use an addition theorem and group algebras to show thatN1(Cp ⊕ Cp) = 2p.

• Show Multiplicity:

N1(Cm⊕Cm) = 2m + N1(Cn⊕Cn) = 2n⇒ N1(Cmn⊕Cmn) = 2mn .

To do so, use the structural description of minimal zero-sumsequences of maximal length over groups of rank two.


How does the typical set of lengths look like?

Theorem (G. + Halter-Koch 2006)

(a) Let S = g1 · . . . · g` be a zero-sum sequence over G

such that

{g1, . . . , g`} ∪ {0} ⊂ G is a subgroup .

Then the set of lengths L(S) is an arithmetical progression

with di�erence 1.

(b) Let R be a ring of integers with class group G.

limx→∞

#{aR | (R :aR) ≤ x and L(a) is an AP as above}#{aR | (R :aR) ≤ x}

= 1 .


Outline


Problems


Sets of Lengths





Distance between factorizations

Let b ∈ H, and let z , z ′ ∈ Z(b) be two factorizations, say

z = w1 · . . . · wn u1 · . . . · uk , z ′ = w1 · . . . · wn v1 · . . . · v`

where all ui , vj ,wk are atoms and ui , vj are pairwise non-associated.

Then

d(z , z ′) = max{k , `} ∈ {0}∪N≥2 the distance between z and z ′.

Note: Let m ∈ N, a = u1 · . . . · uk = v1 · . . . · v`, and study am:

d(

(u1 · . . . · uk)m, (v1 · . . . · v`)m)

= max{km, `m} ≥ 2m ,

but

Z(am) ⊃ {zi = (u1 · . . . · uk)i (v1 · . . . · v`)m−i | i ∈ [0,m]}


Catenary degree

For a ∈ H, let c(a) ∈ N0 ∪ {∞} denote the smallestC ∈ N0 ∪ {∞} with the following property:

For any two factorizations z , z ′ ∈ Z(a), there exists a �nitesequence of factorizations of a

z = z0, z1, . . . , zm = z ′ concatenating z and z ′ in Z(a)

such that

d(zi−1, zi ) ≤ C for all i ∈ [1,m] .

Then

c(H) = sup{c(a) | a ∈ H} is the catenary degree of H .


Arithmetical Properties of the catenary degree

By de�nition, we have• c(H) = 0 ⇐⇒ H is factorial.• If H is not factorial, then 2 + sup∆(H) ≤ c(H). In particular,c(H) = 3 =⇒ all sets of lengths are AP with di�erence 1.

Setc(G ) := c

(B(G )

).

Theorem

• c(G ) ≤ D(G ), and equality holds if and only if

G is either cyclic or an elementary 2-group.

• (G.+Grynkiewicz+Schmid) If D∗(G ) = D(G ), then

k(G ) = 2 + max∆(G ) = c(G ) , where k(G ) =

max{k | u1u2 = v1 · . . . · vk , no lengths between 2 and k}.


Outline


Problems


Sets of Lengths





On not integrally closed noetherian domains

Let R be a noetherian domain with quotient �eld q(R) = K , andR ⊂ K the integral closure of R .

FACTS:

• Every non-zero non-unit of R can be written as a product ofatoms.

• R is Krull with dim(R) = dim(R), andR is Krull if and only if R = R .

• R is a �nitely generated R-module if and only ifthe conductor f = (R :R) = {a ∈ R | aR ⊂ R} 6= {0}.

• Example: Non-principal orders in algebraic number �elds:so in quadratic number �elds

R = Z[f ω] ⊂ R = Z[ω] ⊂ K = Q(ω) .


Arithmetic of not integrally closed noetherian domains

Let R be a noetherian domain with f = (R :R) 6= {0}.Suppose that the class group C(R) and R/f are �nite. Then

• The catenary degree c(R) and the set of distances ∆(R) are�nite.

• There is an M ∈ N0 such that every set of lengths is anAAMP with di�erence d ∈ ∆(R) and bound M.

• There is a �niteness criterion for the re�ned elasticities.For non-principal orders it runs as follows:

• One (equivalently, all) invariants ρk(R) are �nite.• For every nonzero p / R there is precisely one P / R with

P ∩ R = p.

METHOD: Study a transfer homomorphism θ : R → B , where Bis a monoid having similar algebraic properties as R , but which issimpler from a combinatorial point of view.

Documents

Arithmetic of Krull monoids ()Ideals = F (spec )); Dedekind implies ’: R !Ideals : a 7!aR is a divisor homomorphism. De nitions Problems Zero-Sums Sets of Lengths Quantitative