Zero-sum sequences over abelian groups and their systems of sets of - - PowerPoint PPT Presentation

• Zero-sum sequences over abelian groups

and their systems of sets of lengths

Alfred Geroldinger Additive Combinatorics 2020 CIRM, Marseilles, September 7 – 11, 2020

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Outline

Zero-sum sequences and sets of lengths Motivation On L(G) On L(G0) Methods

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Zero-Sum Sequences over groups

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

• A sequence S = (g1, . . . , gℓ) over G0: finite, unordered

sequence of terms from G0, repetition allowed. We set |S| = ℓ.

• S has sum zero if σ(S) = g1 + . . . + gℓ = 0.
• S is a minimal zero-sum sequence if σ(S) = g1 + . . . + gℓ = 0

but no proper subsum equals zero. The monoid of zero-sum sequences:

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

monoid with concatenation of sequences as the operation.

• The minimal zero-sum sequences are the atoms (irreducible

elements) of B(G0).

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

The Davenport constant

The Davenport constant D(G0) of G0 is the maximal length of a minimal zero-sum sequence over G0.

• D(G) is the smallest integer ℓ such that every sequence of

length ≥ ℓ has a non-empty subsequence with sum zero.

• D(G) < ∞ if and only if G is finite.

FACT: Let G = Cn1 ⊕ . . . ⊕ Cnr with 1 < n1 | . . . | nr.

• D∗(G) := 1 + r

i=1(ni − 1) ≤ D(G) ≤ |G|.

• D∗(G) = D(G) for p-groups and in case r ≤ 2.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Sets of lengths in monoids

Monoid H: semigroup with identity element 1H.

• If a = u1 · . . . · uk ∈ H, where u1, . . . , uk ∈ H are irreducible,

then k is called the length of the factorization, and

• LH(a) = {k : a has a factorization of length k} ⊆ N0

is the set of lengths of a.

• We set LH(1H) = {0}.
• 1 ∈ L(a) iff a is irreducible iff L(a) = {1}.
• The system of all sets of lengths

L(H) = {L(a): a ∈ H}

• GOAL. We study L(G0) := L
• B(G0)
• the system of sets of lengths of zero-sum sequences over G0.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Outline

Zero-sum sequences and sets of lengths Motivation On L(G) On L(G0) Methods

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Transfer Krull monoids

A monoid H is transfer Krull if there is a subset G0 of an abelian group and a homomorphism θ: H → B(G0) such that

• θ is surjective up to units.
• θ allows to lift factorizations: if θ(a) = BC, then there are

b, c ∈ H such that θ(b) = B, θ(c) = C, and a = bc. Transfer homomorphisms allow to pull back arithmetical properties from B(G0) to H. In particular,

• LH(a) = LB(G0)
• θ(a)
• for all a ∈ H.
• L(H) = L(G0) := L
• B(G0)
• .

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Krull monoids are transfer Krull

A commutative cancellative monoid is Krull if one of the following equivalent conditions hold.

• H is completely integrally closed and satisfies the ACC on

divisorial ideals.

• There is a divisor homomorphism ϕ: H → F(P), where F(P)

is a free abelian monoid with basis P (divisor hom. means that a divides b in H iff ϕ(a) divides ϕ(b) in F(P)).

Lemma (Classic and simple)

Let H be a Krull monoid with divisor theory ϕ: H → F(P), class group G = C(H), and let G0 ⊆ G denote the set of classes containing prime divisors. Then there is a transfer homomorphism θ: H → B(G0).

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Examples of Krull monoids

• 1. Domains. Let D be a commutative integral domain.

(a) If D is a Dedekind domain, then ϕ: D \ {0} → I∗(D), a → aD, is a divisor theory whence D \ {0} is Krull. (b) If D is integrally closed noetherian, then D \ {0} is Krull. (c) D is a Krull domain iff D \ {0} is a Krull monoid.

• 2. Monoids of Modules. Let R be a ring and let C be a small

class of left R-modules that is closed under isomorphism, direct sums, and direct summands. Then C is (gives rise to) an additive monoid, where the operation is the direct sum and the zero module is the identity element. (Facchini 2002) If EndR(M) is semilocal for all modules M in C, then this monoid is a Krull monoid.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Examples of transfer Krull monoids

Transfer Krull domains, that are not Krull, include subclasses of

• Non-commutative Dedekind domains (Baeth, Smertnig)

In particular (Smertnig 2013), classical maximal orders R in central simple algebras over number fields are transfer Krull iff every stably free left R-ideal is free. If this holds, they are transfer Krull over a finite abelian group.

• Noetherian domains that are close to their complete integral

closure.

• Non-cancellative semigroups of modules.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

What subsets G0 can occur?

• Finite and infinite abelian groups G. They occur in

relevant examples stemming from number theory, algebraic geometry; semigroup rings, finitely generated domains.

• Realization Results. For every abelian group G and every

generating subset G0, there is a Krull monoid (easy) and even a Dedekind domain (Claborn’s Realization Theorem) with class group G and G0 being the set of classes containing prime divisors.

• Module Theory gives a wealth of relevant examples of Krull

monoids with finitely generated class groups G and generating subsets G0 (Baeth, Facchini, Prihoda, Wiegand).

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Outline

Zero-sum sequences and sets of lengths Motivation On L(G) On L(G0) Methods

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

First properties of L(G0)

Let S, S1, S2 be zero-sum sequences and let k ∈ N. (a) If S = 0, then L(S) = {1}. (b) If S = (0, . . . , 0)

• k-times

, then L(S) = {k}. (c) If S = (g1, . . . , gk), then #L(S) ≤ k. (d) If S = (0, . . . , 0)S1, then L(S) = k + L(S1). (e) L(S1) + L(S2) ⊆ L(S1S2). (f) If #L(S) > 1, then #L(S . . . S

k-times

) > k.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Some extremal cases

Let G be an abelian group and G0 ⊆ G be a subset.

• L(G) =
• {k}: k ∈ N0
• iff |G| ≤ 2 iff D(G) ≤ 2.
• Conjecture. Every abelian group contains a generating set G1

such that L(G1) =

• {k}: k ∈ N0
• .
• If G0 is finite but not half-factorial, then there are arbitrarily

large sets L ∈ L(G0) and they are well-structured.

• (Kainrath 1999) If G0 contains an infinite subgroup, then every

finite set L ⊆ N≥2 lies in L(G0).

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Characterizations of class groups

Classic Philosophy in Algebraic Number Theory The class group determines the arithmetic. This was turned into results by the machinery of transfer hom’s. Narkiewicz 1974: Inverse problem Does the arithmetic determine the class group?

• First affirmative answers were given in the 1980s.
• BUT: Which arithmetical properties should be used in the

characterization?

• The best investigated properties are sets of lengths.
• Are sets of lengths sufficient to do the characterization ?

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

MAIN PROBLEM I

MAIN PROBLEM I: The Characterization Problem. Let G be a finite abelian group with Davenport constant D(G) ≥ 4, and let G ′ be an abelian group such that L(G) = L(G ′). Are G and G ′ isomorphic?

• Conjecture. YES!

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Characterizing class groups via sets of lengths

Theorem

Let G be a finite abelian group with D(G) ≥ 4, and let G ′ be an abelian group with L(G) = L(G ′). Then G and G ′ are isomorphic in each of the following cases :

• 1. (G. + Schmid, 2019) G = Cn1 ⊕ Cn2, where

n1, n2 ∈ N with n1 | n2 and n1 + n2 > 4.

• 2. (Zhong, 3 papers, 2017-2018) G = C r

n, where

r, n ∈ N satisfy one of the two conditions :

• r ≤ n − 3
• r ≥ n − 1 and n is a prime power.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Some simple observations

• If L(G ′) = L(G), then D(G ′) = D(G).
• If G ′ G, then L(G ′) L(G).
• For every m ≥ 1, there are (up to isomorphism)
• nly finitely many finite abelian groups G

with Davenport constant D(G) = m.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

New strategy

For m ≥ 4, let Ωm be the family of all systems L(G1), . . . , L(Gk) , where G1, . . . , Gk are the groups with Davenport constant equal to m. We say that L(Gi) is

• maximal in Ωm if L(Gi) ⊆ L(Gj) implies Gi = Gj for all j,
• minimal in Ωm if L(Gj) ⊆ L(Gi) implies Gi = Gj for all j, and
• incomparable in Ωm if it is maximal and minimal in Ωm.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

On the incomparabilty of systems of sets of lengths

Theorem (G. + Schmid 2020)

• 1. For m ∈ [4, 6], L(Cm) is minimal in Ωm, L(C m−1

2

) is maximal in Ωm, and L(Cm) L(C m−1

2

).

• 2. For every m ≥ 7, L(Cm) is incomparable in Ωm and

L(C m−1

2

) is incomparable in Ωm.

• 3. For every m ≥ 5, L(C m−4

2

⊕ C4) is maximal in Ωm.

• 4. For every n ≥ 2, L(C2 ⊕ C2n) is maximal in Ω2n+1. Moreover,

L(C2 ⊕ C2n) is minimal among all L(G) in Ω2n+1 stemming from groups G with D(G) = D∗(G).

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Outline

Zero-sum sequences and sets of lengths Motivation On L(G) On L(G0) Methods

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Properties of systems L(H)

Let H be a monoid such that every element has a factorization into irreducibles, and H not a group. Then we have

• L(1H) = {0} and L(u) = {1} for u ∈ H irreducible.

If 1 / ∈ L(a), then L(a) ⊆ N≥2.

• If u ∈ H is irreducible and k ∈ N, then {k} ⊆ L(uk).
• If a, b ∈ H, then L(a) + L(b) ⊆ L(ab).

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

MAIN PROBLEM II

Let L∗ ⊆ Pfin(N0) be a family of finite subsets of N0 such that

• {0}, {1} ∈ L∗ and all other sets of L∗ lie in N≥2.
• For every k ∈ N0 there is L ∈ L∗ with k ∈ L.
• If L1, L2 ∈ L∗, then there is L ∈ L∗ with L1 + L2 ⊆ L.

MAIN PROBLEM II: Do there exist

• a monoid H such that L∗ = L(H), or even
• a subset G0 of an abelian group G such that L∗ = L(G0)?

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

An extremal case

Consider families L∗ of finite subsets of N0 with the following properties:

• {0}, {1} ∈ L∗ and all other sets of L∗ lie in N≥2.
• If L1, L2 ∈ L∗, then L1 + L2 ∈ L∗.

This means that L∗ is additively closed with respect to set addition, whence

• L∗, +
• ⊆
• Pfin(N0), +
• is a subsemigroup.

Are these reasonable arithmetic conditions?

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

When is L(G) additively closed ?

Proposition

• 1. For a finite abelian group G, the following are equivalent.

(a) All sets of lengths in L(G) are APs with difference in ∆∗(G). (b) All sets of lengths in L(G) are APs. (c) The system of sets of lengths L(G) is additively closed. (d) G is cyclic of order |G| ≤ 4 or isomorphic to a subgroup of C 3

2

• r isomorphic to a subgroup of C 2

3 .

• 2. If G is infinite, then L(G) is additively closed.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

A Realization Theorem for L∗

Theorem (G.+Zhong 2020)

Let L∗ be a family of finite subsets of N0 such that

• {0}, {1} ∈ L∗ and all other sets of L∗ lie in N≥2.
• If L1, L2 ∈ L∗, then L1 + L2 ∈ L∗.

Then there is a Krull monoid H such that L∗ = L(H). Moreover, the following are equivalent.

• L∗ has only finitely many indecomposable sets.
• There is a finitely generated Krull monoid H∗ with L∗ = L(H).

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

A Corollary for Dedekind domains

Corollary

Let L∗ be a family of finite subsets of N0 such that

• {0}, {1} ∈ L∗ and all other sets of L∗ lie in N≥2.
• If L1, L2 ∈ L∗, then L1 + L2 ∈ L∗.

Then there is a Dedekind domain D such that L∗ = L(D). Moreover, the following are equivalent.

• L∗ has only finitely many indecomposable sets.
• There is a Dedekind domain D∗ with L∗ = L(D∗) such that

the number of classes of C(D∗) containing nonzero prime ideals is finite. The proof uses Claborn’s Realization Theorem for class groups.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Outline

Zero-sum sequences and sets of lengths Motivation On L(G) On L(G0) Methods

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Sets of Distances

Recall: If L(G) = L(G ′), then D(G) = D(G ′). Why do we have L(Cn) ⊂ L(G) and L(G) ⊂ L(Cn) for all finite abelian groups G with D(G) = D(Cn) = n ≥ 7? METHOD: Study parameters controlling the structure of sets of lengths. The invariant, studied in the previous talk by Zhong, indicates that L(Cn) and L(C n−1

2

) are different from all other finite abelian groups. For a subset G0 ⊆ G, let ∆(G0) denote the set of successive distances occurring in sets of lengths L ∈ L(G0). Problem 1: Characterize G0 such that ∆(G0) = ∅.

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

Sets of minimal distances

Plagne+Schmid 2005 On the maximal cardinality of half-factorial sets in cyclic groups, Math. Annalen 333, 759 – 785. Problem 2: Study the set of minimal distances ∆∗(G) =

• min ∆(G0): G0 ⊆ G, ∆(G0) = ∅
• .

Plagne+Schmid 2020 On congruence half-factorial Krull monoids with cyclic class group, J. Combinatorial Algebra 3, 331 – 400.

• ∆(Cn) = ∆(C n−1

2

) = ∆∗(C n−1

2

) = [1, n − 2].

• ∆∗(Cn) is NOT an interval.
• L(Cn) = L(C n−1

2

).

Zero-sum sequences/sets of lengths Motivation On L(G) On L(G0) Methods

References

• A. Bashir, A. Geroldinger and Q. Zhong, On a zero-sum

problem arising from factorization theory, arxiv.org/abs/2007.10094.

• A. Geroldinger and W.A. Schmid, A characterization of class

groups via sets of lengths, J. Korean Math. Society 56 (2019), 869 – 915.

• A. Geroldinger and W.A. Schmid, On the incomparability of

systems of sets of lengths, arxiv.org/abs/2005.03316.

• A. Geroldinger and Q. Zhong, Factorization theory in

commutative monoids, Semigroup Forum 100 (2020), 22 – 51.

• A. Geroldinger and Q. Zhong, A realization result for systems
• f sets of lengths, arxiv.org/abs/2008.08820.