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Journal of Algebra | 1988

Almost split sequences for abelian group graded rings

Maurice Auslander; Idun Reiten

While existence theorems for almost split sequences have been developed in a wide variety of settings, it is only recently that any attention has been given to the question of the existence of almost split sequences for categories of graded modules. Not only is this an interesting question in its own right, but the close connection between finitely generated graded modules and coherent sheaves on projective varieties also yields information concerning almost split sequences for categories of coherent sheaves. In [4, 51 we dealt with the question for Z-graded rings. In this paper we deal with rings graded by arbitrary abelian groups. We note that Geigle and Lenzing have studied some special two-dimensional situations in connection with their work on the relationship between vector bundles on weighted projective curves and modules over some particular finite dimensional algebras. Our results and method of proof are for the most part graded versions of those given in [3] for Cohen-Macaulay modules over isolated singularities. However, in some instances we need to provide new arguments. In particular, there is an equivalence of functor categories which plays an important role. In Section 1 we develop the necessary preliminaries on graded categories and functors to be able to state and prove this theorem. In Section 2 we apply this result to obtain our existence theorem for almost split sequences for modules graded by an arbitrary abelian group.


Archive | 1995

Representation Theory of Artin Algebras: Index

Maurice Auslander; Idun Reiten; Sverre O. Smalo

1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary algebras 9. Short chains and cycles 10. Stable equivalence 11. Modules determining morphisms.


Archive | 1995

Representation Theory of Artin Algebras: Contents

Maurice Auslander; Idun Reiten; Sverre O. Smalo

1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary algebras 9. Short chains and cycles 10. Stable equivalence 11. Modules determining morphisms.


Mathematica Scandinavica | 1989

Galois actions on rings and finite Galois coverings.

Maurice Auslander; Idun Reiten; Sverre O. Smalo


Canadian Journal of Mathematics | 1979

Almost split sequences whose middle term has at most two indecomposable summands

Maurice Auslander; R. Bautista; María Inés Platzeck; Idun Reiten; Sverre O. Smalo


Journal of Algebra | 1981

Addendum to “almost split sequences in subcategories”☆

Maurice Auslander; Sverre O. Smalo


Archive | 1991

Applications of homologically nite subcategories

Maurice Auslander; Idun Reiten


Archive | 1996

D Tr-periodic modules and functors

Maurice Auslander; Idun Reiten


Archive | 1987

Almost split sequences for ℤ-graded rings

Maurice Auslander; Idun Reiten


Archive | 1979

Almost split sequences whose middle term has at most two summands

Maurice Auslander; R. Bautista; Maria-Ines Platzeck; Idun Reiten; Sverre O. Smal

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Idun Reiten

Norwegian University of Science and Technology

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Sverre O. Smalo

Norwegian University of Science and Technology

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Sverre O. Smal

Norwegian University of Science and Technology

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