Super-stretched and graded countable Cohen–Macaulay type

Stone, Branden

doi:10.1016/j.jalgebra.2013.09.017

Cited by 3 publications

(8 citation statements)

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“…The hypersurfaces (4) and (5) Proof. As shown in [20,Corollary 4.5], the possible h-vectors are (1), (1, n), or (1, n, 1) for some integer n. Combining this with Corollary 3.5, we know that the possible h-vectors of R are the following:…”

Section: Proofmentioning

confidence: 79%

“…Thus there must be two linear forms a, b ∈ m \ m 2 that are basis elements of m/m 2 . By [20,Lemma 4.1], there are uncountably many distinct homogeneous ideals {I α } α∈k in R. In this context, we have that I α = (a + αb)R. Consider the graded indecomposable maximal Cohen-Macaulay modules {R/I α } α∈k . As each of these modules have different annihilators, we know they are not isomorphic.…”

Section: Zero Dimensional Ringsmentioning

confidence: 99%

“…Let (R, m, k) be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay type, that is not Gorenstein and dim R 3. By [20,Proposition 4.6], we know that R must be a domain and have minimal multiplicity. Thus we only have to consider the three classes of rings listed in Section 4.1 in order to obtain the following.…”

Section: 2mentioning

confidence: 99%

“…The countable analog of this fact is still unknown. Using the concept of super-stretched, Conjecture 5.1 was shown to be true for standard graded rings of dimension at most one [20], but the conjecture remains open for higher dimensions. By the results of [14,6] on hypersurfaces, a positive result to Conjecture 5.1 would completely classify Gorenstein rings of graded countable Cohen-Macaulay type.…”

Section: Gorenstein Ringsmentioning

confidence: 99%

“…With these results, we ask the following question. [20], we know that standard graded complete intersection with graded countable Cohen-Macaulay type are either a hypersurface or defined by two quadrics. With this fact, one could use tools from representation theory [18] and the analysis of the h-vector, to show that the ring is indeed a hypersurface no matter what the dimension.…”

Section: Gorenstein Ringsmentioning

confidence: 99%

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Non-Gorenstein Isolated Singularities of Graded Countable Cohen–Macaulay Type

Stone

2014

Connections Between Algebra, Combinatorics, and Geometry

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Abstract. In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay type.

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Section: Proofmentioning

confidence: 79%

Section: Zero Dimensional Ringsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Gorenstein Ringsmentioning

confidence: 99%

Section: Gorenstein Ringsmentioning

confidence: 99%

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Non-Gorenstein Isolated Singularities of Graded Countable Cohen–Macaulay Type

Stone

2014

Connections Between Algebra, Combinatorics, and Geometry

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Maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum

Kobayashi

Lyle

Takahashi

2020

Journal of Pure and Applied Algebra

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We say that a Cohen-Macaulay local ring has finite CM+-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. In this paper, we consider finite CM+-representation type from various points of view, relating it with several conjectures on finite/countable Cohen-Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite CM+-representation type are exactly the local hypersurfaces of countable CM-representation type, that is, the hypersurfaces of type (A∞) and (D∞). We also discuss the closedness and dimension of the singular locus of a Cohen-Macaulay local ring of finite CM+-representation type.2010 Mathematics Subject Classification. 13C60, 13H10, 16G60.This theorem may look technical, but it actually gives rise to a lot of restrictions which having finite CM + -representation type produces, and is used in the later sections. One concrete example where Theorem 5.5 applies is when I = (x) and (0 : x) = (x); see Corollary 5.9. Here we introduce one of the applications of the above theorem. Denote by CM(R) the category of maximal Cohen-Macaulay R-modules, and by D sg (R) the singularity category of R. Theorem 1.5 (Theorem 5.8). Let R be a Cohen-Macaulay local ring of dimension d > 0. Let I be an ideal of R with V(I) ⊆ V(0 : I) such that R/I is maximal Cohen-Macaulay over R.Suppose that R has finite CM + -representation type. Then one must have d = 1. If I n = 0 for some integer n > 0, then CM(R) has dimension at most n − 1 in the sense of [18]. If R is Gorenstein, then R is a hypersurface and D sg (R) has dimension at most n − 1 in the sense of [23].There are folklore conjectures that a Gorenstein local ring of countable CM-representation type is a hypersurface, and that, for a Cohen-Macaulay local ring R of countable CM-representation type, CM(R) has dimension at most one. The above theorem gives partial answers to the variants of these folklore conjectures for finite CM + -representation type.

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