ω-k- torsionfree modules and ω-left approximation dimension

The notion of ω-k-torsionfree modules with respect to a bimodule ω is introduced, which is characterized in terms of left add_R ω-approximations. The notion of ω-left approximation dimension is introduced, and the forms ofk-syzygy modules beingk-torsionfree modules are described. Dedicated to Professor Liu Shaoxue on his 70th birthday.     

On Modules with Finite Exchange Property

The purpose of this paper is to answer an open problem proposed by Matlis on modules with the finite exchange property. The problem is: Let {M _i | i ∈ I} be an indexed family of indecomposable injective modules and M a direct sum of M _i , i.e. M = ⊕{M _i | i ∈ I} and N a summand of M. Is N a direct sum of indecomposable injective modules, i.e. N = ⊕{M _j |j ∈ J, J ? I}? The answer to this problem is affirmative for module M with finite exchange property.     

Homological Computations in PBW Modules

In this paper the Poincaré–Birkhoff–Witt (PBW) rings are characterized. Gröbner bases techniques are also developed for these rings. An explicit presentation of Ext ⁱ (M,N) is provided when N is a centralizing bimodule.     

Conjugation in Representations of the Zassenhaus Algebra

Let F be an algebracially closed field of characteristic p > 2, and L be the p ⁿ -dimensional Zassenhaus algebra with the maximal invariant subalgebra L ₀ and the standard filtration {L _i }| ^pn−2 _{i =−1}. Then the number of isomorphism classes of simple L-modules is equal to that of simple L ₀-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p ⁿ . Received May 10, 1999, Accepted December 8, 1999     

Topological modules of continuous homomorphisms

In this paper, we consider the notion of module homomorphisms in the general topological module setting and establish their linearity and continuity under some suitable conditions. We also introduce the strict and uniform topologies on the modules of continuous linear homomorphisms and study their various properties.     

Abelian group pairs having a trivial coGalois group

Torsion-free covers are considered for objects in the category q ₂. Objects in the category q ₂ are just maps in R-Mod. For R = ℤ, we find necessary and sufficient conditions for the coGalois group G(A → B), associated to a torsion-free cover, to be trivial for an object A → B in q ₂. Our results generalize those of E. Enochs and J. Rado for abelian groups.     

Sum formulas for reductive algebraic groups

Let V be a Weyl module either for a reductive algebraic group G or for the corresponding quantum group Uq. If G is defined over a field of positive characteristic p, respectively if q is a primitive lth root of unity (in an arbitrary field) then V has a Jantzen filtration V=V⁰⊃V¹⊃?⊃Vr=0. The sum of the positive terms in this filtration satisfies a well-known sum formula.If T denotes a tilting module either for G or Uq then we can similarly filter the space HomG(V,T), respectively HomUq(V,T) and there is a sum formula for the positive terms here as well.We give an easy and unified proof of these two (equivalent) sum formulas. Our approach is based on an Euler type identity which we show holds without any restrictions on p or l. In particular, we get rid of previous such restrictions in the tilting module case.     

Phase Spaces and Deformation Theory

In the papers (Laudal in Contemporary Mathematics, vol. 391, [2005]; Geometry of time-spaces, Report No. 03, [2006/2007]), we introduced the notion of (non-commutative) phase algebras (spaces) Ph ⁿ (A), n=0,1,…,∞ associated to any associative algebra A (space), defined over a field k. The purpose of this paper is to study this construction in some more detail. This seems to give us a possible framework for the study of non-commutative partial differential equations. We refer to the paper (Laudal in Phase spaces and deformation theory, Report No. 09, [2006/2007]), for the applications to non-commutative deformation theory, Massey products and for the construction of the versal family of families of modules. See also (Laudal in Homology, Homotopy, Appl. 4:357–396, [2002]; Proceedings of NATO Advanced Research Workshop, Computational Commutative and Non-Commutative Algebraic Geometry, [2004]).