The Auslander generators of the exterior algebra in two variables
We compute a complete set of nonisomorphic minimal Auslander generators for the exterior algebra in two variables.
The category of abelian Hopf algebras
The composite of irreducible morphisms in regular components
We study when the composite of n irreducible morphisms between modules in a regular component of the Auslander-Reiten quiver is non-zero and lies in the n+1-th power of the radical ℜ of the module category. We prove that in this case such a composite belongs to . We apply these results to characterize those string algebras having n irreducible morphisms between band modules such that their composite is a non-zero morphism in .
The dimension of the derived category of elliptic curves and tubular weighted projective lines
We show that the dimension of the derived category of an elliptic curve or a tubular weighted projective line is one. We give explicit generators realizing this number, and show that they are in a certain sense minimal.
The duality theorem for twisted smash products of Hopf algebras and its applications
Let denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for , and as an application we establish the relationship between the homological dimensions of and A if H and its dual H* are both semisimple.
The existence of relative pure injective envelopes
Let 𝓢 be a class of finitely presented R-modules such that R∈ 𝓢 and 𝓢 has a subset 𝓢* with the property that for any U∈ 𝓢 there is a U*∈ 𝓢* with U* ≅ U. We show that the class of 𝓢-pure injective R-modules is preenveloping. As an application, we deduce that the left global 𝓢-pure projective dimension of R is equal to its left global 𝓢-pure injective dimension. As our main result, we prove that, in fact, the class of 𝓢-pure injective R-modules is enveloping.
The flat dimensions of injective modules.
The Gelfand-Kirillov Dimension of a Tensor Product.
The general structure of inverse polynomial modules
In this paper we compute injective, projective and flat dimensions of inverse polynomial modules as -modules. We also generalize Hom and Ext functors of inverse polynomial modules to any submonoid but we show Tor functor of inverse polynomial modules can be generalized only for a symmetric submonoid.
The global homological dimension of semi-trival extensions of rings.
The Global Homological Dimension of Some Algebras of Differential Operators.
The homological dimension of a torsion-free abelian group of finite rank as a module over its ring of endomorphisms
The Homological Dimension of an Abelian Group as a Module over its Ring of Endomorphisms. III.
The Homological Dimension of an Abelian Group as a Module over Its Ring of Endomorphisms. II.
The representation dimension of domestic weakly symmetric algebras
Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras...
The vanishing of self-extensions over n-symmetric algebras of quasitilted type
A ring Λ satisfies the Generalized Auslander-Reiten Condition ( ) if for each Λ-module M with for all i > n the projective dimension of M is at most n. We prove that this condition is satisfied by all n-symmetric algebras of quasitilted type.
Tilting Modules of Finite Projective Dimension.
Tilting theory and quasitilted algebras.
Totally reflexive modules with respect to a semidualizing bimodule
Let and be two associative rings, let be a semidualizing -bimodule. We introduce and investigate properties of the totally reflexive module with respect to and we give a characterization of the class of the totally -reflexive modules over any ring . Moreover, we show that the totally -reflexive module with finite projective dimension is exactly the finitely generated projective right -module. We then study the relations between the class of totally reflexive modules and the Bass class...
Towards a theory of Bass numbers with application to Gorenstein algebras
The notion of Gorenstein rings in the commutative ring theory is generalized to that of Noetherian algebras which are not necessarily commutative. We faithfully follow in the steps of the commutative case: Gorenstein algebras will be defined using the notion of Cousin complexes developed by R. Y. Sharp [Sh1]. One of the goals of the present paper is the characterization of Gorenstein algebras in terms of Bass numbers. The commutative theory of Bass numbers turns out to carry over with no extra changes....