Representation type of commutative Noetherian rings. III. Global wildness and tameness

Klingler, Lee; Levy, Lawrence S.

doi:10.1090/memo/0832

Cited by 15 publications

(30 citation statements)

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“…Of course it is impossible to do so if R is a principal ideal ring. More generally, recall from [29,30,31] that a local ring (R, m, k) is Dedekind-like provided R is reduced and one-dimensional, the integral closure R is generated by at most two elements as an R-module, and m is the Jacobson radical of R. In a long and difficult paper [30] Levy and Klinger classify the indecomposable finitely generated modules over most Dedekind-like rings. There is one exceptional case where the classification has not yet been worked out, namely, where R is a local domain whose residue field is purely inseparable of degree two over k. We will call these Dedekind-like rings exceptional.…”

Section: Modules With Torsionmentioning

confidence: 99%

Direct-sum behavior of modules over one-dimensional rings

Karr

Wiegand²

2010

Commutative Algebra

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Let R be a reduced, one-dimensional Noetherian local ring whose integral closure R is finitely generated over R. Since R is a direct product of finitely many principal ideal domains (one for each minimal prime ideal of R), the indecomposable finitely generated R-modules are easily described, and every finitely generated R-module is uniquely a direct sum of indecomposable modules. In this article we will see how little of this good behavior trickles down to R. Indeed, there are relatively few situations where one can describe all of the indecomposable R-modules, or even the torsion-free ones. Moreover, a given finitely generated module can have many different representations as a direct sum of indecomposable modules. Finite Cohen-Macaulay typeIf R is a one-dimensional reduced Noetherian local ring, the maximal CohenMacaulay R-modules (those with depth 1) are exactly the non-zero finitely generated torsion-free modules. One says that R has finite Cohen-Macaulay type provided there are, up to isomorphism, only finitely many indecomposable maximal Cohen-Macaulay modules. The following theorem classifies these rings:

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Section: Modules With Torsionmentioning

confidence: 99%

Direct-sum behavior of modules over one-dimensional rings

Karr

Wiegand²

2010

Commutative Algebra

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“…For consistency with the notation in [KL1]- [KL3], we shall always write maps on the right for the remainder of §2. The proof of Theorem 1.2 divides naturally into the two cases of Proposition 2.2.…”

Section: Constructions In Dimension Onementioning

confidence: 99%

“…This structure is so simple that one is tempted to try to generalize Steinitz's result to a larger class of commutative rings. Indeed, in a recent series of papers [KL1], [KL2], [KL3] L. Klingler and L. Levy presented a classification, up to isomorphism, of all finitely generated modules over a class of commutative rings they call "Dedekind-like".…”

Section: §1 Introductionmentioning

confidence: 99%

Indecomposable modules of large rank over Cohen-Macaulay local rings

Häßler

Karr

Klingler

et al. 2008

Trans. Amer. Math. Soc.

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Abstract. A commutative Noetherian local ring (R, m, k) is called Dedekindlike provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. If M is an indecomposable finitely generated module over a Dedekind-like ring R, and if P is a minimal prime ideal of R, it follows from a classification theorem due to L. Klingler and L. Levy that M P must be free of rank 0, 1 or 2.Now suppose (R, m, k) is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let P 1 , . . . , P t be the minimal prime ideals of R. The main theorem in the paper asserts that, for each non-zero t-tuple (n 1 , . . . , n t ) of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated R-modules M satisfyingIn 1911, E. Steinitz determined the structure of all finitely generated modules over Dedekind domains. This structure is so simple that one is tempted to try to generalize Steinitz's result to a larger class of commutative rings. Indeed, in a recent series of papers [KL1], [KL2], [KL3] L. Klingler and L. Levy presented a classification, up to isomorphism, of all finitely generated modules over a class of commutative rings they call "Dedekind-like".We recall that a commutative Noetherian local ring (R, m, k) is Dedekind-like [KL1, Definition 2.5] provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. (In [KL2, (1.1.3)] a further requirement is imposed: If R/m is a field, then it is a separable extension of k. Klingler and Levy prove their classification theorem only under this additional hypothesis. In the present paper, however, we do not require that Dedekind-like rings satisfy this separability condition.) Although Dedekind-like rings are very close to their normalizations, their module structure is much more complicated than that of Dedekind domains. Klingler and Levy dash any hope of a further extension of their classification theorem by showing that, if R is not a homomorphic image of a Dedekind-like ring or a special type

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“…Let R be the m-adic completion of R. All hypotheses on R transfer to R (cf. [KL3,Lemma 11.8] We now assume, by way of contradiction, that D is a complete local Dedekind-like ring and σ : D R is a surjective ring homomorphism. Suppose first that R is reduced.…”

Section: Finding a Suitable Finite-length Modulementioning

confidence: 99%

Big Indecomposable Mixed Modules over Hypersurface Singularities

Häßler¹,

Wiegand²

2006

Abelian Groups, Rings, Modules, and Homological Algebra

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IntroductionThis research began as an effort to determine exactly which one-dimensional local rings have indecomposable finitely generated modules of arbitrarily large constant rank. The approach, which uses a new construction of indecomposable modules via the bimodule structure on certain Ext groups, turned out to be effective mainly for hypersurface singularities. The argument was eventually replaced by a direct, computational approach [HKKW], which applies to all one-dimensional Cohen-Macaulay local rings.In this paper we resurrect the Ext argument to build indecomposable modules of large rank over hypersurface singularities of any dimension d ≥ 1. The main point of the construction is that, modulo an indecomposable finite-length part, the modules constructed are maximal Cohen-Macaulay modules. Thus, even when there are no indecomposable maximal Cohen-Macaulay modules of large rank, we can build short exact sequencesin which T and X are indecomposable, T has finite length, and F is maximal CohenMacaulay of arbitrarily large constant rank. The main result (Theorem 2.3) on building indecomposables is quite general, and it is likely that there are other contexts where it will prove useful.In order to state our main application, we establish some terminology. Let k be a field. By a hypersurface singularity we mean a commutative Noetherian local ring (R, m, k) whose m-adic completion R is isomorphic to S/(f ), where (S, n, k) is a complete regular local ring and f is a non-zero element of n 2 . A Noetherian local ring (R, m, k) is Dedekind-like [KL1, Definition 2.5] provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. (Examples include discrete valuation rings and rings such as k [[x, y]

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Representation type of commutative Noetherian rings. III. Global wildness and tameness

Cited by 15 publications

References 0 publications

Direct-sum behavior of modules over one-dimensional rings

Direct-sum behavior of modules over one-dimensional rings

Indecomposable modules of large rank over Cohen-Macaulay local rings

Big Indecomposable Mixed Modules over Hypersurface Singularities

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