Existence of Auslander-Reiten sequences in subcategories

This paper studies the existence of Auslander-Reiten sequences in subcategories of mod(Λ), where Λ is a finite dimensional algebra over a field. The two main theorems give necessary and sufficient conditions for the existence of Auslander-Reiten sequences in subcategories.

Theorem. LetMbe a subcategory ofmod(Λ)closed under extensions and direct summands, and letMbe an indecomposable module inMsuch thatfor someinM. Then the following are equivalent.

(i)

has an-precover in the stable category,

(ii)

There is an Auslander-Reiten sequence0→X→Y→M→0inM.

     

Normal sequences over finite abelian groups

In this note, we obtain the structure of short normal sequences over a finite abelian p-group or a finite abelian group of rank two, thus answering positively a conjecture of Gao and Zhuang for various groups. The results obtained here improve all known results on this conjecture.     

On the index of minimal zero-sum sequences over finite cyclic groups

Let G be a cyclic group of order n?2 and a sequence over G. We say that S is a zero-sum sequence if and that S is a minimal zero-sum sequence if S is a zero-sum sequence and S contains no proper zero-sum sequence.The notion of the index of a minimal zero-sum sequence (see Definition 1.1) in G has been recently addressed in the mathematical literature. Let l(G) be the smallest integer t∈N such that every minimal zero-sum sequence S over G with length |S|?t satisfies index(S)=1. In this paper, we first prove that for n?8. Secondly, we obtain a new result about the multiplicity and the order of elements in long zero-sumfree sequences.     

Auslander-Reiten sequences under base field extension

We investigate the behaviour of Auslander-Reiten sequences of modules over a finite dimensional algebra over a field under base field extension. It is proved that an Auslander-Reiten sequence splits into a direct sum of Auslander-Reiten sequences provided the extension is separable in the sense of MacLane.

     

Long zero-free sequences in finite cyclic groups

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than n/2 in the additive group Z_n of integers modulo n. The main result states that for each zero-free sequence  of length ?>n/2 in Z_n there is an integer g coprime to n such that if denotes the least positive integer in the congruence class ga_i (modulo n), then . The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than n/2, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.     

On finite tetraprimary groups

A three-player game is considered in which the first and second players have dynamic superiority over the third player. Two fixed time points are specified. The game ends if either the first player captures the third player at the first time point, or the second player captures the third player at the second time point. We analyze a situation when the initial positions in the game are such that neither the first nor the second player alone can capture the third player at the specified points of time. We propose sufficient conditions on the parameters of the game under which, for given initial states of the players, the first and second players by applying some controls can guarantee that one of them will meet the third player at the prescribed moment. Simulation results for a model example are also presented.     

Lie algebras determined by finite valued Auslander-Reiten quivers

Let г denote a connected valued Auslander-Reiten quiver, let ℒ(γ) denote the free abelian group generated by the vertex setγ ₀ and let ℒ(Γ) be the universal cover ofг with fundamental groupG. It is proved that whenγ is a finite connected valued Auslander-Reiten quiver,ℒ(γ) is a Lie subalgebra ofℋ(г), and is just the “orbit” Lie algebra ℒ( )/G, where ℋ (г)₁ is the degenerate Hall algebra ofг and ℒ( )/G is the “orbit” Lie algebra induced by .