掲載種別：研究論文（学術雑誌）   出版者・発行元：Elsevier BV  

DOI： 10.1016/j.jalgebra.2022.04.021

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掲載種別：研究論文（学術雑誌）   出版者・発行元：Cambridge University Press (CUP)  

Abstract

We study powers of binomial edge ideals associated with closed and block graphs.

DOI： 10.1017/nmj.2021.1

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掲載種別：研究論文（学術雑誌）   出版者・発行元：Springer Science and Business Media LLC  

DOI： 10.1007/s40687-022-00326-2

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その他リンク： https://link.springer.com/article/10.1007/s40687-022-00326-2/fulltext.html

掲載種別：研究論文（学術雑誌）   出版者・発行元：Springer Science and Business Media LLC  

DOI： 10.1007/s40306-021-00441-2

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その他リンク： https://link.springer.com/article/10.1007/s40306-021-00441-2/fulltext.html

掲載種別：研究論文（学術雑誌）   出版者・発行元：Informa UK Limited  

DOI： 10.1080/00927872.2021.1917590

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掲載種別：研究論文（学術雑誌）  

© 2020 Elsevier Inc. We give a complete characterization of graphs whose binomial edge ideal is licci. An important tool is a new general upper bound for the regularity of binomial edge ideals.

DOI： 10.1016/j.jcta.2020.105278

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掲載種別：研究論文（国際会議プロシーディングス）  

© 2020, Springer Nature Switzerland AG. Via computer-aided classification we show that the following three conditions are equivalent for level* squarefree monomial ideals I with codimension 3, with Cohen-Macaulay type 2 and with is licci, (2) the twisted conormal module of I is Cohen-Macaulay, (3) is Cohen-Macaulay, where S is a polynomial ring over a field of characteristic 0 and is its graded maximal ideal.

DOI： 10.1007/978-3-030-52111-0_10

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掲載種別：研究論文（学術雑誌）  

© 2019 Elsevier Inc. In this paper, we prove that the t-th ordinary and/or symbolic power of a Stanley-Reisner ideal is level for some positive integer t≥3 if and only if IΔ is a complete intersection and equi-generated. For t=2, we give a characterization of level property of the second symbolic power IΔ(2) when Δ is a matroid complex of dimension one.

DOI： 10.1016/j.jalgebra.2019.05.044

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掲載種別：研究論文（学術雑誌）  

© 2019, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. We classify licci Gorenstein squarefree monomial ideals I with dim S/I ≤ 4, where S is a polynomial ring.

DOI： 10.1007/s40306-019-00339-0

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掲載種別：研究論文（学術雑誌）  

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. We study the relationship between depth and regularity of a homogeneous ideal I and those of (I, f) and I : f, where f is a linear form or a monomial. Our results have several interesting consequences on depth and regularity of edge ideals of hypergraphs and of powers of ideals.

DOI： 10.1007/s10801-018-0811-9

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掲載種別：研究論文（学術雑誌）  

© 2019 by the authors. We show that Cohen-Macaulay and (S2) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal I such that S/I2 satisfies the Serre condition (S2), but is not Cohen-Macaulay.

DOI： 10.3390/math7080684

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掲載種別：研究論文（学術雑誌）  

© 2017 by The Editorial Board of the Nagoya Mathematical Journal. A very well-covered graph is an unmixed graph whose covering number is half of the number of vertices. We construct an explicit minimal free resolution of the cover ideal of a Cohen-Macaulay very well-covered graph. Using this resolution, we characterize the projective dimension of the edge ideal of a very well-covered graph in terms of a pairwise -disjoint set of complete bipartite subgraphs of the graph. We also show nondecreasing property of the projective dimension of symbolic powers of the edge ideal of a very well-covered graph with respect to the exponents.

DOI： 10.1017/nmj.2017.7

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掲載種別：研究論文（学術雑誌）  

© 2016 Elsevier Inc. We give a bound for n0(I) from which the depth function depthR/I(n) of the quotient ring by symbolic powers of a squarefree monomial ideal I stabilizes. In the unmixed codimension two case we show that depthR/I(n) is a non-increasing function of n and use this property to provide a sharp bound for n0(I).

DOI： 10.1016/j.jalgebra.2016.10.036

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掲載種別：研究論文（学術雑誌）  

© 2015, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore. In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I of a polynomial ring S is equal to the projective dimension of S/I when arithdeg I−indeg I=2 and I has a linear resolution.

DOI： 10.1007/s40306-015-0136-x

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掲載種別：研究論文（学術雑誌）  

© 2014, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore. In this paper, we pose the following conjecture and give a positive answer to the case dimΔ≤2: Let Δ be a (d−1)-dimensional simplicial complex on [n]. Fix an integer ℓ with 0≤ℓ≤n−d−1. If e(K[Δ])≤(ℓ+1)d−ℓ and βℓ,ℓ+d(K[Δ])=0, then reg K[Δ]≤d−1. Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.

DOI： 10.1007/s40306-014-0103-y

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掲載種別：研究論文（学術雑誌）  

© 2014 Glasgow Mathematical Journal Trust. Let R = k[x 1,..,x n], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We show that R/It(Cn) is Sr if and only if it is Cohen-Macaulay or n n-t-1 r+3. In addition, we prove that R/It(Cn) is Gorenstein if and only if n = t or 2t + 1. Also, we determine all ordinary and symbolic powers of It(Cn) which are Cohen-Macaulay. Finally, we prove that It(Cn) has a linear resolution if and only if t ≥ n-2.

DOI： 10.1017/S0017089514000111

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掲載種別：研究論文（学術雑誌）  

© 2014 Elsevier Inc. We prove that a squarefree monomial ideal of height 3 whose quotient ring is Gorenstein is a set-theoretic complete intersection.

DOI： 10.1016/j.jalgebra.2014.09.005

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掲載種別：研究論文（学術雑誌）  

Let K{double-struck} be a field, and let R = K{double-struck}[x1,.., xn] be the polynomial ring over K{double-struck} in n indeterminates x1,.., xn. Let G be a graph with vertex-set {x1,.., xn}, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J(k) and J[k], respectively. In this paper, we give necessary and sufficient conditions for R/Jk, R/J (k), and R/J [k] to be Cohen-Macaulay. We also study the limit behavior of the depths of these rings. © 2015 Copyright Taylor & Francis Group, LLC.

DOI： 10.1080/00927872.2014.897550

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掲載種別：研究論文（学術雑誌）  

We present combinatorial characterizations for the associated primes of the second power of squarefree monomial ideals and criteria for this power to have positive depth or depth greater than one. © 2013 Elsevier B.V.

DOI： 10.1016/j.jpaa.2013.11.008

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掲載種別：研究論文（学術雑誌）  

© 2014 Rocky Mountain Mathematics Consortium. The problem of finding a characterization of Cohen-Macaulay simplicial complexes has been studied intensively by many authors. There are several attempts at this problem available for some special classes of simplicial complexes satisfying some technical conditions. This paper is a survey, with some new results, of some of these developments. The new results about simplicial complexes with Serre's condition are an analogue of the known results for Cohen-Macaulay simplicial complexes.

DOI： 10.1216/JCA-2014-6-4-455

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掲載種別：研究論文（学術雑誌）  

We study squarefree monomial ideals which are in the linkage class of a complete intersection (we call such an ideal licci). Firstly we classify all licci edge ideals. Secondly we prove that any Cohen-Macaulay almost complete intersection squarefree monomial ideal is licci. Thirdly we characterize licci squarefree monomial ideals of deviation 2 in terms of hypergraphs associated to the ideals as an application of the classification in Kimura, Terai and Yoshida (2009) [15]. © 2013 Elsevier Inc.

DOI： 10.1016/j.jalgebra.2013.06.001

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掲載種別：研究論文（学術雑誌）  

We prove that the binomial arithmetical rank of the edge ideal of a forest coincides with its big height. © 2013 American Mathematical Society.

DOI： 10.1090/S0002-9939-2013-11473-5

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掲載種別：研究論文（学術雑誌）  

We classify the unmixed squarefree lexsegment ideals and determine those which are Cohen-Macaulay. © 2012 Copyright Taylor and Francis Group, LLC.

DOI： 10.1080/00927872.2011.605409

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掲載種別：研究論文（学術雑誌）  

Let I be a squarefree monomial ideal of a polynomial ring S. In this article, we prove that the arithmetical rank of I is equal to the projective dimension of S/I when one of the following conditions is satisfied: (1) μ(I) ≤5; (2) arithdeg I ≤ 4. © 2012 Copyright Taylor and Francis Group, LLC.

DOI： 10.1080/00927872.2011.602781

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掲載種別：研究論文（学術雑誌）  

We prove that for m≥3, the symbolic power Iδ(m) of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex δ is a matroid. Similarly, the ordinary power Iδm is Cohen-Macaulay for some m≥3 if and only if Iδ is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals. © 2011 Elsevier Inc.

DOI： 10.1016/j.aim.2011.10.004

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掲載種別：研究論文（学術雑誌）  

Let S=K[x1,..., xn] be a polynomial ring over a field K. Let I(G)⊆S denote the edge ideal of a graph G. We show that the ℓth symbolic power I(G)(ℓ) is a Cohen-Macaulay ideal (i.e., S/I(G)(ℓ) is Cohen-Macaulay) for some integer ℓ≥3 if and only if G is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers I(G)(ℓ) are Cohen-Macaulay ideals. Similarly, we characterize graphs G for which S/I(G)(ℓ) has (FLC).As an application, we show that an edge ideal I(G) is complete intersection provided that S/I(G)ℓ is Cohen-Macaulay for some integer ℓ≥3. This strengthens the main theorem in Crupi et al. (2010) [3]. © 2011 Elsevier Inc.

DOI： 10.1016/j.jalgebra.2011.09.007

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掲載種別：研究論文（学術雑誌）  

A graph is called very well-covered if it is unmixed without isolated vertices such that the cardinality of each minimal vertex cover is half the number of vertices. We first prove that a very well-covered graph is Cohen-Macaulay if and only if it is vertex decomposable. Next, we show that the Castelnuovo-Mumford regularity of the quotient ring of the edge ideal of a very well-covered graph is equal to the maximum number of pairwise 3-disjoint edges. © 2011 Elsevier B.V.

DOI： 10.1016/j.jpaa.2011.02.005

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掲載種別：研究論文（学術雑誌）  

We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition Sr. In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially Sr if and only if its pure i-skeleton is Sr for all i. For r = 2, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially S r if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first r steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially Sr cycles showing that the only sequentially S2 cycles are odd cycles and, for r ≥ 3, no cycle is sequentially Sr with the exception of cycles of length 3 and 5. We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially Sr graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially S2. We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially S2. Finally, we propose some questions. © 2010 American Mathematical Society.

DOI： 10.1090/S0002-9939-2010-10646-9

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掲載種別：研究論文（学術雑誌）  

The lemma given by Schmitt and Vogel is an important tool in the study of the arithmetical rank of squarefree monomial ideals. In this paper, we give a Schmitt-Vogel type lemma for reductions as an analogous result. © 2011 Springer Basel AG.

DOI： 10.1007/s00013-011-0256-z

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掲載種別：研究論文（学術雑誌）  

We consider a class of graphs G such that the height of the edge ideal I(G) is half of the number #V (G) of the vertices. We give Cohen-Macaulay criteria for such graphs. © 2011 by The Editorial Board of the Nagoya Mathematical Journal.

DOI： 10.1215/00277630-2010-018

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掲載種別：研究論文（学術雑誌）  

We show that the Stanley-Reisner ideal of the one-dimensional simplicial complex whose diagram is an n-gon is always a set-theoretic complete intersection in any positive characteristic. © Taylor & Francis Group, LLC.

DOI： 10.1080/00927871003597634

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掲載種別：研究論文（学術雑誌）  

Let R = k[x1, . . . , xn], where k is a field. The path ideal (of length t ≥ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cnbe an n-cycle. We determine when I t(Cn) is unmixed. Moreover, We show that R/I t(Cn) is sequentially Cohen-Macaulay if and only if n = t or t + 1 or 2t + 1.

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掲載種別：研究論文（学術雑誌）  

In this paper, we study several properties of the second power (formula) of a Stanley-Reisner ideal I∆ of any dimension. As the main result, we prove that S/I∆ is Goren-stein whenever (formula) is Cohen-Macaulay over any field K. Moreover, we give a criterion for the second symbolic power of I∆ to satisfy (S2) and to coincide with the ordinary power, respectively. Finally, we provide new examples of Stanley-Reisner ideals whose second powers are Cohen-Macaulay. © 2011, Rocky Mountain Mathematics Consortium. All rights reserved.

DOI： 10.1216/JCA-2011-3-3-405

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掲載種別：研究論文（学術雑誌）  

When a cone is added to a simplicial complex Δ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex Δ′. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of Δ′ equals the projective dimension of the Stanley-Reisner ring of Δ′ if the corresponding equality holds for Δ. © Taylor & Francis Group, LLC.

DOI： 10.1080/00927870903236186

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掲載種別：研究論文（学術雑誌）  

Let I ⊂ S = K[x1,...,xn] be a lexsegment edge ideal or the Alexander dual of such an ideal. In both cases it turns out that the arithmetical rank of I is equal to the projective dimension of S/I.

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掲載種別：研究論文（学術雑誌）  

Let R = K[x1, . . ., xn] be a polynomial ring over a field K. Let I = I(G) ⊆ R be the edge ideal of a graph G. We show that I is complete intersection if R/Il is Cohen-Macaulay for some l ≥ height I. This strengthens the Cowsik-Nori theorem in the case of edge ideals. © Taylor & Francis Group, LLC.

DOI： 10.1080/00927870903114995

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掲載種別：研究論文（学術雑誌）  

We study h-vectors of simplicial complexes which satisfy Serre's condition (Sr). Let r be a positive integer. We say that a simplicial complex △ satisfies Serre's condition (Sr) if H̃ i(lk△ (F);K) = 0 for all F ∈ △ and for all i < min{r-1, dim lk△ (F)}, where lk△ (F) is the link of △ with respect to F and where H̃i(△;K) is the reduced homology groups of △ over a field K. The main result of this paper is that if △ satisfies Serre's condition (Sr) then (i) hk(△) is non-negative for k = 0, 1, . . ., r and (ii) ∑k≥r hk(△) is non-negative. © International Press 2009.

DOI： 10.4310/MRL.2009.v16.n6.a10

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掲載種別：研究論文（学術雑誌）  

In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/I in the following cases: (a) I is an almost complete intersection; (b) arithdeg∈I=reg∈I; (c) arithdeg∈I=indeg∈I+1. We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c). © 2008 Springer Science+Business Media, LLC.

DOI： 10.1007/s10801-008-0142-3

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掲載種別：研究論文（学術雑誌）  

In this paper, we prove that the Stanley-Reisner ideal of any connected simplicial complex of dimension ≥ 2 that is locally complete intersection is a complete intersection ideal. As an application, we show that the Stanley-Reisner ideal whose powers are Buchsbaum is a complete intersection ideal. © 2010 University of Illinois.

DOI： 10.1215/ijm/1266934785

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掲載種別：研究論文（学術雑誌）  

Let be a (d-1)-dimensional simplicial complex on the vertex set V={1, 2, n}. In this article, using Alexander duality, we prove that the Stanley-Reisner ring k[Δ] is Cohen-Macaulay if it satisfies Serre's condition (S2) and the multiplicity e(k[Δ]) is "sufficiently large", that is, [image omitted]. We also prove that if e(k[Δ])3d-2 and the graded Betti number 2, d+2(k[Δ]) vanishes, then the Castelnuovo-Mumford regularity regk[Δ] is less than d.

DOI： 10.1080/00927870701716124

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掲載種別：研究論文（学術雑誌）  

We prove that Stanley-Reisner rings having sufficiently large multiplicities are Cohen-Macaulay using Alexander duality. © 2005.

DOI： 10.1016/j.jalgebra.2005.10.011

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掲載種別：研究論文（学術雑誌）  

First, we give a new criterion for Buchsbaum Stanley-Reisner rings to have linear resolutions. Next, we prove that every ( d - 1)-dimensional complex Δ of initial degree d is contained in the same dimensional Cohen-Macaulay complex whose ( d - 1)th reduced homology is isomorphic to that of Δ. We call such a simplicial complex a Cohen-Macaulay cover of Δ. And we also show that all the intermediate complexes between Δ and its Cohen-Macaulay cover are Buchsbaum provided that Δ is Buchsbaum. As an application, we determine the h -vectors of the 3-dimensional Buchsbaum Stanley-Reisner rings with initial degree 3.

DOI： 10.1080/00927870600651638

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掲載種別：研究論文（学術雑誌）  

In this paper, we study Buchsbaum Stanley-Reisner rings with linear free resolution. We introduce the notion of Buchsbaum Stanley-Reisner rings with minimal multiplicity of initial degree q, which extends the notion of Buchsbaum rings with minimal multiplicity defined by Goto. As an application, we give many examples of non-Cohen-Macaulay Buchsbaum StanleyReisner rings with linear resolution. © 2005 American Mathematical Society.

DOI： 10.1090/S0002-9939-05-08176-1

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掲載種別：研究論文（学術雑誌）  

Algebraic and combinatorial properties of a monomial ideal and its radical are compared. © Birkhäuser Verlag, Basel 2005.

DOI： 10.1007/s00013-005-1385-z

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記述言語：英語   掲載種別：研究論文（学術雑誌）  

DOI： 10.1016/j.jsc.2004.04.001

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掲載種別：研究論文（学術雑誌）  

DOI： 10.1007/BF03322817

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掲載種別：研究論文（学術雑誌）  

Let P(v, d) be a stacked d-polytope with v vertices, Δ(P(v,d)) the boundary complex of P(v,d), and k[Δ(P(v,d))] = A/IΔ(P(v,d)) the Stanley-Reisner ring of Δ(P(v,d)) over a field k. We compute the Betti numbers which appear in a minimal free resolution of k[Δ(P(v,d))] over A, and show that every Betti number depends only on v and d and is independent of the base field k. We also show that the Betti number sequences above are unimodal.

DOI： 10.1007/bf02678204

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掲載種別：研究論文（学術雑誌）  

A technique of minimal free resolutions of Stanley-Reisner rings enables us to show the following two results: (1) The 1-skeleton of a simplicial (d - 1)-sphere is d-connected, which was first proved by Barnette; (2) The comparability graph of a non-planar distributive lattice of rank d - 1 is d-connected.

DOI： 10.1023/a:1008648302195

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掲載種別：研究論文（学術雑誌）  

DOI： 10.1006/aima.1996.0086

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掲載種別：研究論文（学術雑誌）  

We study the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ring k[Δ] = A/IΔ of a simplicial complex Δ over a field k. It is known that the second Betti number of k[Δ] is independent of the base field k. We show that, when the ideal IΔ is generated by square-free monomials of degree two, the third and fourth Betti numbers are also independent of k. On the other hand, we prove that, if the geometric realization of Δ is homeomorphic to either the 3-sphere or the 3-ball, then all the Betti numbers of k[Δ] are independent of the base field k.

DOI： 10.1016/S0012-365X(96)83021-4

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掲載種別：研究論文（学術雑誌）  

We give a combinatorial formula for the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ring k[Δ(P)] = A/IΔ(P), of the boundary complex Δ(P) of an odd-dimensional cyclic polytope P over a field k. A corollary to the formula is that the Betti number sequence of k[Δ(P)] is unimodal and does not depend on the base field k.

DOI： 10.1007/BF02711496

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掲載種別：研究論文（学術雑誌）  

We give a necessary condition for a sequence of integers to be the h-vector of a Buchsbaum complex (or equivalently a Buchsbaum Stanley-Reisner ring). We construct 3-dimensional Buchsbaum Stanley-Reisner rings with depth 2 which give lower bounds of the h-vectors among those of the Buchsbaum Stanley-Reisner rings with the above conditions. © 1996 by the University of Notre Dame. All rights reserved.

DOI： 10.14492/hokmj/1351516714

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掲載種別：研究論文（学術雑誌）  

First in order to investigate deformation of ASLs, we define the moduli space of ASLs on a given poset. And we give an inequality between the depth of general ASLs and that of the corresponding Stanley-Reisner ring, which includes the fundamental theorem on ASLs by De Concini, Eisenbud and Procesi (1982). Secondly we give a counter-example of the following conjecture of Hibi: If there exist an ASL on a finite poset H over a field k which is an integral domain then H is Cohen-Macaulay over k, i.e., the Stanlay-Reisner ring k[H] of H is Cohen-Macaulay. © 1994.

DOI： 10.1016/0022-4049(94)90120-1

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掲載種別：研究論文（学術雑誌）  

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