Stanley-Reisner ideals whose powers  have finite length cohomologies

Gotō, Shirō; Takayama, Yukihide

doi:10.1090/s0002-9939-07-08795-3

Cited by 12 publications

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“…Similarly, if t = f r for some r > m, then T + e t ∈ W i+1,F , and z T +e t = 0 by (5). Finally, for any t ∈ F, s(T + e t ) = s(T ) = F, and so s(T +e t ),s(T ) is the identity map.…”

Section: Lemma 52 Allows Us To Get a Handle On Hmentioning

confidence: 92%

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Face rings of simplicial complexes with singularities

2010

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The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of Cohen-Macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an r -dimensional complex , it is equivalent to nonsingularity of in dimension r − c; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules. E.123 858 E. Miller et al.

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“…Similarly, if t = f r for some r > m, then T + e t ∈ W i+1,F , and z T +e t = 0 by (5). Finally, for any t ∈ F, s(T + e t ) = s(T ) = F, and so s(T +e t ),s(T ) is the identity map.…”

Section: Lemma 52 Allows Us To Get a Handle On Hmentioning

confidence: 92%

“…Connections between face rings and modules with finite local cohomology have been studied in [5,21].…”

Section: Remark 22mentioning

confidence: 99%

Face rings of simplicial complexes with singularities

2010

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“…If we move on to the higher powers of I(G), then we can graph-theoretically characterize G such that I(G) m is Cohen-Macaulay (or Buchsbaum, or generalized CohenMacaulay) for some m 3 (and for all m 1) (see [4,15,19]). For the second power, we proved that I(G) 2 is Cohen-Macaulay if and only if G is a triangle-free graph in W 2 (see [10]).…”

Section: Introductionmentioning

confidence: 99%

“…This problem was considered for certain classes of graphs (see [5,6,9,10]). Generally, we cannot read off the Cohen-Macaulay and Gorenstein properties of G just from its structure because these properties in fact depend on the characteristic of the base field K (see [20, Exercise 5.3.31] and [10, Proposition 2.1]).If we move on to the higher powers of I(G), then we can graph-theoretically characterize G such that I(G) m is Cohen-Macaulay (or Buchsbaum, or generalized CohenMacaulay) for some m 3 (and for all m 1) (see [4,15,19] …”

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confidence: 99%

Buchsbaumness of the second powers of edge ideals

Hoang

Trung

2018

J. Algebra Appl.

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Abstract. We graph-theoretically characterize the class of graphs G such that I(G) 2 are Buchsbaum. IntroductionThroughout this paper let G = (V (G), E(G)) be a finite simple graph without isolated vertices. An independent set in G is a set of vertices no two of which are adjacent to each other. The size of the largest independent set, denoted by α(G), is called the independence number of G. A graph is called well-covered if every maximal independent set has the same size. A well-covered graph G is a member of the class W 2 if the remove any vertex of G leaves a well-covered graph with the same independence number as G (see e.g. [14]).Let R = K[x 1 , . . . , x n ] be a polynomial ring of n variables over a given field K. Let G be a simple graph on the vertex set V (G) = {x 1 , . . . , x n }. We associate to the graph G a quadratic squarefree monomial idealwhich is called the edge ideal of G. We say that G is Cohen-Macaulay (resp. Gorenstein) if I(G) is a Cohen-Macaulay (resp. Gorenstein) ideal. It is known that G is well-covered whenever it is Cohen-Macaulay (see e.g. [20, Proposition 6.1.21]) and G is in W 2 whenever it is Gorenstein (see e.g. [10, Lemma 2.5]). It is a wide open problem to characterize graph-theoretically the Cohen-Macaulay (resp. Gorenstein) graphs. This problem was considered for certain classes of graphs (see [5,6,9,10]). Generally, we cannot read off the Cohen-Macaulay and Gorenstein properties of G just from its structure because these properties in fact depend on the characteristic of the base field K (see [20, Exercise 5.3.31] and [10, Proposition 2.1]).If we move on to the higher powers of I(G), then we can graph-theoretically characterize G such that I(G) m is Cohen-Macaulay (or Buchsbaum, or generalized CohenMacaulay) for some m 3 (and for all m 1) (see [4,15,19]

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“…It is known that, when ∆ = core∆, K[∆] is a GCI if and only if S/I ∆ has FLC for all ≥ 1. (See [4,Theorem 2.5]). For the notations and definitions on simplicial complexes and Stanley-Reisner rings, we follow [1] and throughout this paper we will always consider simplicial complexes ∆ over the vertex set [n] = {1, .…”

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confidence: 99%

Generalized complete intersections with linear resolutions

Okudaira

Takayama

2008

Arch. Math.

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We determine the simplicial complexes ∆ whose Stanley-Reisner ideals I∆ have the following property: for all n ≥ 1 the powers I n ∆ have linear resolutions and finite length local cohomologies. Mathematics Subject Classification (2000). Primary 13F55; Secondary 13D02, 13D45.Introduction. Let S = K[X 1 , . . . , X n ] be a standard graded polynomial ring over a field K and m = (X 1 , . . . , X n ). Recall that a graded ideal I ⊂ S is called generalized Cohen-Macaulay or simply FLC (finite local cohomology) when the local cohomology H i m (S/I) has finite length for all i < dim S/I. We find many examples of such ideals in algebraic geometry: the defining ideals of Cohen-Macaulay equidimensional projective schemes over a field are all FLC ideals. However, as far as the authors are concerned, we do not know very much about FLC monomial ideals. The second author gave a combinatorial characterization of FLC monomial ideals [8] as an extension of the well known case of squarefree monomial ideals, i.e., Buchsbaum Stanley-Reisner ideals. On the other hand, the notion of generalized complete intersections (GCI) has been introduced in [4]. A GCI I ⊂ S is a squarefree monomial ideal such that the Stanley-Reisner ring S/I is a complete intersection over the punctured spectrum. All the powers I n (n ≥ 1) of a GCI I ⊂ S are FLC, and a combinatorial characterization of GCI has been given.

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Stanley-Reisner ideals whose powers have finite length cohomologies

Abstract: Abstract. We introduce a class of Stanley-Reisner ideals called a generalized complete intersection, which is characterized by the property that all the residue class rings of powers of the ideal have FLC. We also give a combinatorial characterization of such ideals.

Cited by 12 publications

References 5 publications

Face rings of simplicial complexes with singularities

Face rings of simplicial complexes with singularities

Buchsbaumness of the second powers of edge ideals

Generalized complete intersections with linear resolutions

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