Stable module theory

Auslander, Maurice; Bridger, Mark

doi:10.1090/memo/0094

Cited by 779 publications

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“…As it is shown in the paper of Auslander and Bridger [1], the G-dimension enjoys several good properties. We recall some of them from [1] for the later use.…”

Section: Preliminaries For G-dimensionmentioning

confidence: 89%

“…Since ΩM is also an indecomposable module of G-dimension zero, it follows from Step 7 that ΩM is minimally generated by b elements in (ΩM ) 1 and that there is a minimal free cover as a graded R-module: R(−1) b → ΩM . Thus one can continue this procedure to get the result stated in Step 8.…”

Section: Preliminaries For G-dimensionmentioning

confidence: 99%

“…The notion of G-dimension of finitely generated modules are introduced by Auslander and Bridger [1]. Although various properties concerning G-dimension has been known, it still seems to lack references which show us many examples of modules of G-dimension zero.…”

Section: Introductionmentioning

confidence: 99%

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Modules of G-Dimension Zero over Local Rings with the Cube of Maximal Ideal Being Zero

Yoshino

2003

Commutative Algebra, Singularities and Computer Algebra

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Let (R, m) be a commutative Noetherian local ring with m 3 = (0). We give a condition for R to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero with parameters in an open subset of projective space. We shall finally show that the subcategory consisting of modules of G-dimension zero over R is not necessarily a contravariantly finite subcategory in the category of finitely generated R-modules.

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“…As it is shown in the paper of Auslander and Bridger [1], the G-dimension enjoys several good properties. We recall some of them from [1] for the later use.…”

Section: Preliminaries For G-dimensionmentioning

confidence: 89%

Section: Preliminaries For G-dimensionmentioning

confidence: 99%

See 1 more Smart Citation

Modules of G-Dimension Zero over Local Rings with the Cube of Maximal Ideal Being Zero

Yoshino

2003

Commutative Algebra, Singularities and Computer Algebra

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“…(1) There is an R-module M with G-dim R M < ∞ = pd R M , and m is decomposable; (2) R is Gorenstein, and m is decomposable; (3) There are a complete regular local ring S of dimension two and a regular system of parameters x, y of S such that R ∼ = S/(xy).…”

Section: Introductionmentioning

confidence: 99%

Direct Summands of Syzygy Modules of the Residue Class Field

Takahashi

2008

Nagoya Mathematical Journal

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Abstract. Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite Gdimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely. IntroductionThroughout the present paper, we assume that all rings are commutative Noetherian local rings and all modules are finitely generated modules.G-dimension is a homological invariant of a module which has been introduced by Auslander [1]. This invariant is an analogue of projective dimension. Whereas the finiteness of projective dimension characterizes the regular property of the base ring, the finiteness of G-dimension characterizes the Gorenstein property of the base ring. To be precise, any module over a Gorenstein local ring has finite G-dimension, and a local ring with residue class field of finite G-dimension is Gorenstein. Gdimension shares a lot of properties with projective dimension. For example, it also satisfies an Auslander-Buchsbaum-type equality, which is called the AuslanderBridger formula.Dutta [9] proved the following theorem in his research into the homological conjectures: Theorem 1.0.1 (Dutta). Let (R, m, k) be a local ring. Suppose that the nth syzygy module of k has a non-zero direct summand of finite projective dimension for some n ≥ 0. Then R is regular.Since G-dimension is similar to projective dimension, this theorem naturally leads us to the following question: Question 1.0.2. Let (R, m, k) be a local ring. Suppose that the nth syzygy module of k has a non-zero direct summand of finite G-dimension for some n ≥ 0. Then is R Gorenstein?It is obviously seen from the indecomposability of k that this question is true if n = 0. Hence this question is worth considering just in the case where n ≥ 1.We are able to answer in this paper that the above question is true if n ≤ 2. Furthermore, as the theorems below say, we can even determine the structure of a ring satisfying the assumption of the above question for n = 1, 2.2000 Mathematics Subject Classification. Primary 13D02; Secondary 13D05, 13H10.

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“…For the basics on the Auslander-Bridger transpose Tr:R-Ind -> Ind-R resp. Tr: Ind -R -> R-Ind for a semiperfect ring R we refer the reader to [4] 2. Strong preinjective partitions and strong preprojective partitions…”

Section: Introductionmentioning

confidence: 99%

Strong preinjective partitions and representation type of Artinian rings

Zimmermann-Huisgen¹

1990

Proc. Amer. Math. Soc.

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Abstract.It is shown that for every ring of left pure global dimension zero (i.e., for every ring all of whose left modules are direct sums of finitely generated modules), the finitely generated left modules can be grouped to a unique "strong preinjective partition" while the finitely presented right modules possess a "strong preprojective partition"; these strong partitions are upgraded versions of the partitions introduced by Auslander and Smal0 for Artin algebras. One direct consequence is that a ring of left pure global dimension zero has finite representation type if and only if there exist sufficiently many almost split maps among its finitely generated left modules. This provides a very elementary proof for Auslander's theorem saying that for Artin algebras vanishing of the left pure global dimension is equivalent to finiteness of the representation type.

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Stable module theory

Cited by 779 publications

References 0 publications

Modules of G-Dimension Zero over Local Rings with the Cube of Maximal Ideal Being Zero

Modules of G-Dimension Zero over Local Rings with the Cube of Maximal Ideal Being Zero

Direct Summands of Syzygy Modules of the Residue Class Field

Strong preinjective partitions and representation type of Artinian rings

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