On the radical of a monomial ideal

Herzog, Jürgen; Takayama, Yukihide; Terai, Naoki

doi:10.1007/s00013-005-1385-z

Cited by 67 publications

(58 citation statements)

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“…Assume that R is normal and I ⊂ R is a (not necessarily monomial) ideal. In §6, generalizing a result of Herzog, Takayama and Terai [9] [20] showed that the range…”

Section: Introductionmentioning

confidence: 76%

See 1 more Smart Citation

Notes onC-Graded Modules Over an Affine Semigroup RingK[C]

Yanagawa

2008

Communications in Algebra

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Abstract. Let C ⊂ N d be an affine semigroup, and R = K[C] its semigroup ring. This paper is a collection of various results on "C-graded" R-modules M = c∈C M c , especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radical monomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a topological property of the "geometric realization" of the cell complex associated with I. Moreover, we can give a squarefree modules/constructible sheaves version of this result. We also show that if R is normal and I ⊂ R is a Cohen-Macaulay monomial ideal then √ I is Cohen-Macaulay again. IntroductionFirst, we fix the notation used throughout this paper. Let C ⊂ Z d ⊂ R d be an affine semigroup (i.e., C is a finitely generated additive submonoid of Z d ), andOf course, R = c∈C Kx c is a Z d -graded ring. We say a Z d -graded ideal of R is a monomial ideal. Let *Mod R be the category of Z d -graded R-modules and their degree preserving R-homomorphisms, and *mod R its full subcategory consisting of finitely generated modules. As usual, for M ∈ *Mod R and a ∈ Z d , M a denotes the degree a component of M, and M(a) denotes the shifted module of M with M(a) b = M a+b . We say M ∈ *Mod R is C-graded, if M a = 0 for all a ∈ C. A monomial ideal I ⊂ R and the quotient ring R/I are C-graded modules. Let *mod C R be the full subcategory of *mod R consisting of C-graded modules.Miller [14] proved that *mod C R has enough injectives and any object has a minimal injective resolution in this category, which is unique up to isomorphism and has finite length. This resolution is called a minimal irreducible resolution, since an indecomposable injective in *mod C R corresponds to a monomial irreducible ideal.In §2, under the assumption that R is Cohen-Macaulay and simplicial, we show that information on M ∈ *mod C R such as depth and Cohen-Macaulay property can be read off from numerical invariants of the minimal irreducible resolution of M (something analogous to "Bass numbers"). One might think these results should be 2000 Mathematics Subject Classification. Primary 13D02; Secondary 13H10, 13F55, 13D45.

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“…Assume that R is normal and I ⊂ R is a (not necessarily monomial) ideal. In §6, generalizing a result of Herzog, Takayama and Terai [9] [20] showed that the range…”

Section: Introductionmentioning

confidence: 76%

“…But the next theorem states that if √ I is a monomial ideal then such an example does not exist. When R is a polynomial ring, this result was obtained in [9]. To extend Proposition 7.4 to semigroup rings, we have to introduce the notion supp + (u) for u ∈ R d .…”

Section: Ideals Whose Radicals Are Monomial Idealsmentioning

confidence: 98%

Notes onC-Graded Modules Over an Affine Semigroup RingK[C]

Yanagawa

2008

Communications in Algebra

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“…Thus the questions discussed in this paper can be specified as follows: For which squarefree Cohen-Macaulay monomial ideals does there exist at least one nontrivial Cohen-Macaulay modification, or do exist infinitely many nontrivial CohenMacaulay modifications, and for which Cohen-Macaulay monomial ideals I are all monomial ideals J ∈ G I or J ∈ F I CohenMacaulay? Related questions have been studied in [4,1].…”

Section: Macaulay Monomial Ideal In Smentioning

confidence: 99%

Cohen–Macaulay monomial ideals with given radical

Ahmad

Naeem

2010

Journal of Pure and Applied Algebra

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“…It is well known that depth S/I ≤ depth S/ √ I (see the proof of [8,Theorem 2.6]) and equivalently depth I ≤ depth √ I. The first inequality holds also for sdepth, that is sdepth S/I ≤ sdepth S/ √ I (see [2,Theorem 1]).…”

Section: Introductionmentioning

confidence: 99%