The Chern–Galois character

International Journal of Mathematics and Mathematical Sciences, 2005

We show that every finitely generated left R-module in the Auslander class over an nperfect ring R having a dualizing module and admitting a Matlis dualizing module has a Gorenstein projective cover.

Algebras and Representation Theory, 2019

In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra Λ is representation-finite if and only if Λ is right n-Nakayama for some positive integer n. We classify hereditary right n-Nakayama algebras. We also define right n-coNakayama algebras and show that an artin algebra Λ is right n-coNakayama if and only if Λ is left n-Nakayama. We then study right 2-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences over a right 2-Nakayama algebra. We end by classifying finite dimensional right 2-Nakayama algebras in terms of their quivers with relations. 1. Introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as an R-module. Given an artin algebra Λ, it is a quite natural question to ask for the classification of all the indecomposable finitely generated right Λ-modules. Only for few classes of algebras such a classification is known, one of the first such class were the Nakayama algebras. A Nakayama algebra Λ is an algebra such that the indecomposable projective right Λ-modules as well as the indecomposable injective right Λ-modules are uniserial. This then implies that all the indecomposable right Λmodules are uniserial. Nakayama algebras were studied by Tadasi Nakayama who called them generalized uniserial rings [9, 10]. A right Λ-module M is called uniserial if it has a unique composition series. Uniserial modules are the simplest indecomposables and this makes it interesting to understand their role in the category mod(Λ) of finitely generated right Λ-modules. An artin algebra Λ is said to be representation-finite, provided there are only finitely many isomorphism classes of indecomposable right Λ-modules. In representation theory, representation-finite algebras are of particular importance since in this case one has a complete combinatorial description of the module category in terms of the Auslander-Reiten quiver. The class of Nakayama algebras is one of the fundamental classes of representation-finite algebras whose representation theory is completely understood. In this paper we introduce the notion of n-factor serial modules. We say that a nonuniserial right Λ-module M of length l is n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. In some sense, n is an invariant that measures how far M is from being uniserial. We say that an artin algebra Λ is right n-Nakayama if every finitely generated indecomposable right Λ-module is i-factor serial for some 1 i n

Transactions of the American Mathematical Society, 1984

Journal of Algebra, 2003

2006

I would like to start by thanking my tutor, Arne B. Sletsjøe, without whose support, encouragement and contagious enthusiasm this thesis would not have been finished. Thanks also to Astrid, Øyvind and Marit for keeping me sane. And last but not least, thanks to my family. So in the noncommutative case, we are still interested in the structure of the category of finitely generated graded A-modules modulo torsion. Our aim is to show that the set P, which in the commutative case corresponds to the set of closed points in X = Proj (A), can be parametrised by the set F of finite dimensional simple A-modules, and by the set C of 1-critical modules that possess such simples as quotients. We then outline a method of finding the finite dimensional simple quotients of certain 1-critical modules, namely the point modules, and finally demonstrate how this method works in some explicit examples. I Theory Chapter 1 Graded Algebras Let us start by giving the definitions of the objects we want to study, and the restrictions we put on them: Let k be a field, not necessarily algebraically closed, and A a k-algebra. Given a (not necessarily commutative) semigroup (G, +), the algebra A is G-graded if we have a direct decomposition of the underlying additive group A = n∈G A n , such that the ring multiplication maps A m ⊗ A n into A m+n , ∀m, n ∈ G. A (left) G-graded module is a module M over a G-graded algebra A, that can be decomposed into M = n∈G M n , such that the action of A on M maps A m ⊗ M n into M m+n , ∀m, n ∈ G. A submodule N ⊂ M is G-graded if N = n∈G (N ∩ M n). A graded module homomorphism f : M → N of degree m between two graded A-modules is an A-module homomorphism satisfying f (M n) ⊆ N m+n for all n. Elements in A n and M n are called homogeneous elements of degree n.

Communications in Algebra, 2006

Generalising the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring C for which PA is finitely generated and projective and the evaluation map µC : Hom C (P, C) ⊗S P → C is an isomorphism (of corings) where S = End C (P ). It was observed that for such comodules the functors HomA(P, −) ⊗S P and − ⊗A C from the category of right A-modules to the category of right C-comodules are isomorphic. In this note we call modules P with this property Galois comodules without requiring PA to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. These comodules are close to being generators and have some common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions. 1 in [4, 18.26] that this condition implies that the functors Hom A (P, −) ⊗ S P and − ⊗ A C from the right A-modules to the right C-comodules are isomorphic.

Journal of Pure and Applied Algebra, 1991

Algebra Universalis, 2010

Journal of Pure and Applied Algebra, 1984

Let A be a finite-dimensional basic connected associative algebra over an algebraicall\ closed field, and T(A) =A K DA its trivial extension by its minimal injective cogenerator. M.'tb proie that T(A) is representation-finite of Cartan class d if and only if A is an iterated tilted algebra of Dynkin class d. The proof also yields a construction procedure for iterated tilted algebra? ot Dynkin type.

Journal of Pure and Applied Algebra, 2002

We investigate internal groupoids and pseudogroupoids in varieties of universal algebras, and we give a new description of internal groupoids in congruence modular varieties. We then prove that in any congruence modular variety an algebraically central extension is categorically central. The converse implication being already known, it follows that there is a perfect agreement between these two notions in any congruence modular variety. This theorem extends various partial results in this direction proved, so far, for-groups, for Maltsev varieties and for semi-abelian categories.