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 ( G 0 ) Methods Outline Zero-sum sequences and sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods Zero-Sum Sequences over groups Let G be an additive abelian group and G 0 ⊆ G be a subset. • A sequence S = ( g 1 , . . . , g ℓ ) over G 0 : finite, unordered sequence of terms from G 0 , repetition allowed. We set | S | = ℓ . • S has sum zero if σ ( S ) = g 1 + . . . + g ℓ = 0. • S is a minimal zero-sum sequence if σ ( S ) = g 1 + . . . + g ℓ = 0 but no proper subsum equals zero. The monoid of zero-sum sequences : • The set B ( G 0 ) of zero-sum sequences over G 0 is a monoid with concatenation of sequences as the operation. • The minimal zero-sum sequences are the atoms (irreducible elements) of B ( G 0 ) .

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods The Davenport constant The Davenport constant D ( G 0 ) of G 0 is the maximal length of a minimal zero-sum sequence over G 0 . • 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 = C n 1 ⊕ . . . ⊕ C n r with 1 < n 1 | . . . | n r . • D ∗ ( G ) := 1 + � r i = 1 ( n i − 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 ( G 0 ) Methods Sets of lengths in monoids Monoid H : semigroup with identity element 1 H . • If a = u 1 · . . . · u k ∈ H , where u 1 , . . . , u k ∈ H are irreducible, then k is called the length of the factorization, and • L H ( a ) = { k : a has a factorization of length k } ⊆ N 0 is the set of lengths of a . • We set L H ( 1 H ) = { 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 ( G 0 ) := L B ( G 0 ) the system of sets of lengths of zero-sum sequences over G 0 .

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods Outline Zero-sum sequences and sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods Transfer Krull monoids A monoid H is transfer Krull if there is a subset G 0 of an abelian group and a homomorphism θ : H → B ( G 0 ) 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 ( G 0 ) to H . In particular, � � • L H ( a ) = L B ( G 0 ) θ ( a ) for all a ∈ H . � � • L ( H ) = L ( G 0 ) := L B ( G 0 ) .

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) 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 G 0 ⊆ G denote the set of classes containing prime divisors. Then there is a transfer homomorphism θ : H → B ( G 0 ) .

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) 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 End R ( 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 ( G 0 ) 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 ( G 0 ) Methods What subsets G 0 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 G 0 , there is a Krull monoid (easy) and even a Dedekind domain (Claborn’s Realization Theorem) with class group G and G 0 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 G 0 (Baeth, Facchini, Prihoda, Wiegand).

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods Outline Zero-sum sequences and sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods First properties of L ( G 0 ) Let S , S 1 , S 2 be zero-sum sequences and let k ∈ N . (a) If S = 0, then L ( S ) = { 1 } . , then L ( S ) = { k } . (b) If S = ( 0 , . . . , 0 ) � �� � k -times (c) If S = ( g 1 , . . . , g k ) , then # L ( S ) ≤ k . (d) If S = ( 0 , . . . , 0 ) S 1 , then L ( S ) = k + L ( S 1 ) . (e) L ( S 1 ) + L ( S 2 ) ⊆ L ( S 1 S 2 ) . (f) If # L ( S ) > 1, then # L ( S . . . S ) > k . � �� � k -times

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods Some extremal cases Let G be an abelian group and G 0 ⊆ G be a subset. � � • L ( G ) = { k } : k ∈ N 0 iff | G | ≤ 2 iff D ( G ) ≤ 2. • Conjecture. Every abelian group contains a generating set G 1 � � such that L ( G 1 ) = { k } : k ∈ N 0 . • If G 0 is finite but not half-factorial, then there are arbitrarily large sets L ∈ L ( G 0 ) and they are well-structured. • (Kainrath 1999) If G 0 contains an infinite subgroup, then every finite set L ⊆ N ≥ 2 lies in L ( G 0 ) .

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) 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 ( G 0 ) 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 ( G 0 ) 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 = C n 1 ⊕ C n 2 , where n 1 , n 2 ∈ N with n 1 | n 2 and n 1 + n 2 > 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 ( G 0 ) 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) only finitely many finite abelian groups G with Davenport constant D ( G ) = m .

Zero-sum sequences/sets of lengths Motivation On L ( G ) On L ( G 0 ) Methods New strategy For m ≥ 4, let Ω m be the family of all systems L ( G 1 ) , . . . , L ( G k ) , where G 1 , . . . , G k are the groups with Davenport constant equal to m . We say that L ( G i ) is • maximal in Ω m if L ( G i ) ⊆ L ( G j ) implies G i = G j for all j , • minimal in Ω m if L ( G j ) ⊆ L ( G i ) implies G i = G j for all j , and • incomparable in Ω m if it is maximal and minimal in Ω m .