Manuscript Drafts of Other Tales c.1813–1818

Crabbe, George; Dalrymple-Champneys, Norma; Pollard, Arthur

doi:10.1093/oseo/instance.00040046

Cited by 1 publication

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“…If R has finite Cohen-Macaulay type, such a classification has been worked out by Nicholas Baeth and Melissa Luckas in [1] and [2]. At the other extreme, when each analytic branch has infinite Cohen-Macaulay type, Andrew Crabbe and Silvia Saccon [8] have a result similar to Theorem 4.7 above, from which one can decode the structure of C(R). The intermediate case, e.g.,…”

Section: Is Not a Discrete Valuation Ring And Qmentioning

confidence: 96%

“…If, now, ϕ is an isomorphism, we see from (8) that α has to be invertible. If, in addition, t = u, the third equation in (11) gives a contradiction, since the left side is invertible and the right side is not.…”

Section: Is a Prinicpal Ideal Ring And Dimmentioning

confidence: 99%

“…Squaring the first equation in (8), and comparing "1" and "a" terms, we see that α 2 = α and γ = αγ + γα. But equation (11) says that αH = Hα, and it follows that α is in k [H], which is a local ring.…”

Section: Is a Prinicpal Ideal Ring And Dimmentioning

confidence: 99%

“…, where R has infinite CohenMacaulay type but at least one branch has finite Cohen-Macaulay type, is discussed in [8], but much less is known about the possible ranks of the indecomposables in this case.…”

Section: Is Not a Discrete Valuation Ring And Qmentioning

confidence: 99%

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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: Is Not a Discrete Valuation Ring And Qmentioning

confidence: 96%

Section: Is a Prinicpal Ideal Ring And Dimmentioning

confidence: 99%