Flat cyclic modules

Necessary and sufficient conditions are established for cyclic -modules to be flat, and rings over which any flat cyclic -module is free are investigated.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 245–250, August, 1970.The author wishes to thank L. A. Skornyakov for his advice and suggestions for improving this article.     

Artinian level modules of embedding dimension two

We prove that a sequence of positive integers (h₀,h₁,…,h_c) is the Hilbert function of an artinian level module of embedding dimension two if and only if h_i−1−2h_i+h_i+1≤0 for all 0≤i≤c, where we assume that h₋₁=h_c+1=0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.     

More about the embedding of a cyclic extension into a cyclic one

Relationship between the concordance condition and the solvability of the problem of embedding a cyclic Galois extension into another cyclic Galois extension is investigated. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 293–299.     

On embedding infinite cyclic covers in compact 3-manifolds

We show that there are knots whose infinite cyclic covers do not embed in any compact 3-manifold, answering a question of Jiang et al. (2006).     

Trivial source modules in blocks with cyclic defect groups

We shall present a method to get trivial source modules easily just by looking at values of ordinary characters at non-identity p-elements in finite groups instead of doing huge calculation. The method is only for a case where defect groups are cyclic. Nevertheless, it works well at least when we want to prove Broué’s abelian defect group conjecture for blocks which have elementary abelian defect groups of order p ².     

Scott modules in finite groups with cyclic Sylow p-subgroups

We present a simple method for determining Scott modules of finite groups using just the Brauer tree of the principal block and the values of ordinary characters at non-identity p-elements. Previously, determining Scott modules often involved long calculations. The method is applicable to the case where finite groups have cyclic Sylow p-subgroups. The structure of a Scott module is closely related to the length of the shortest path from the vertex labeled by the trivial ordinary character to the exceptional vertex in the Brauer tree. We classify the structures of the Scott modules and the ordinary characters they afford, by the length of the associated path in the Brauer tree.     

Essential normality of quasi-homogeneous quotient modules over the polydisc

In this paper, we characterize the essential normality of quasi-homogeneous quotient Hardy modules over the polydisc by giving a complete criterion in terms of partially maximal ideals. Then we generalize this result to the weighted Bergman modules.     

On the universally solvable embedding problem with cyclic kernel

It is proved that the universally solvable embedding problem with cyclic kernel is semidirect. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 173–179.     

Degenerations for indecomposable modules in almost cyclic coherent Auslander-Reiten components

We give a necessary and sufficient condition for the existence of degeneration M≤_degN for arbitrary modules M, N of the same dimension from the additive category of a generalized standard almost cyclic coherent component of the Auslander-Reiten quiver of finite-dimensional algebra.