Faculty Profiles - TERAI Naoki

Papers - TERAI Naoki

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On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley–Reisner ideals of small height Reviewed

M. R. Pournaki, M. Poursoltani, N. Terai, S. Yassemi

Journal of Algebra 632 751 - 782 2023.10

Publishing type：Research paper (scientific journal)

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector contains important information on the Serre's condition (Sr) and the CMt property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d−1)-dimensional simplicial complexes with codimension two which are (Sd−3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h-vectors and give a negative answer to a known question regarding these vectors.

DOI： 10.1016/j.jalgebra.2023.05.031

Licci level Stanley-Reisner ideals with height three Invited Reviewed

Giancarlo Rinaldo, Naoki Terai

Sao Paulo Journal of Mathematical Sciences 17 ( 1 ) 345 - 386 2023.6

Authorship：Corresponding author Publishing type：Research paper (scientific journal)

Using a computer we classify licci level* squarefree monomial ideals I with codimension 3 and with dim S/ I≤ 4. We also show the following two conditions are equivalent: (1) ISm is licci, (2) the twisted conormal module of I is Cohen-Macaulay, where S is a polynomial ring and m is its graded maximal ideal.

DOI： 10.1007/s40863-022-00326-8

Simplicial Complexes Satisfying Serre's Condition versus the Ones Which Are Cohen--Macaulay in a Fixed Codimension Reviewed

M. R. Pournaki, M. Poursoltani, N. Terai, S. Yassemi

SIAM Journal on Discrete Mathematics 36 ( 4 ) 2506 - 2522 2022.12

A note on monomial ideals which are Cohen–Macaulay in a fixed codimension Reviewed

M. R. Pournaki, K. Shibata, N. Terai, S. Yassemi

Communications in Algebra 50 ( 11 ) 4988 - 4996 2022.11

Sequentially Cohen–Macaulay binomial edge ideals of closed graphs Reviewed

Viviana Ene, Giancarlo Rinaldo, Naoki Terai

Research in the Mathematical Sciences 9 ( 3 ) 2022.9

Very well-covered graphs and local cohomology of their residue rings by the edge ideals Reviewed

K. Kimura, M.R. Pournaki, N. Terai, S. Yassemi

Journal of Algebra 606 1 - 18 2022.9

POWERS OF BINOMIAL EDGE IDEALS WITH QUADRATIC GRÖBNER BASES Reviewed

VIVIANA ENE, GIANCARLO RINALDO, NAOKI TERAI

Nagoya Mathematical Journal 246 233 - 255 2022.6

A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra Reviewed

K. Kimura, M. R. Pournaki, S. A. Seyed Fakhari, N. Terai, S. Yassemi

Research in the Mathematical Sciences 9 ( 2 ) 2022.6

A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension Reviewed

M.R. Pournaki, M. Poursoltani, N. Terai, S. Yassemi

Acta Mathematica Vietnamica 47 ( 1 ) 181 - 196 2022.3

Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs Reviewed

Seyed Amin Seyed Fakhari, Kosuke Shibata, Naoki Terai, Siamak Yassemi

Communications in Algebra 1 - 13 2021.4

Licci binomial edge ideals Invited Reviewed

Viviana Ene, Giancarlo Rinaldo, Naoki Terai

Journal of Combinatorial Theory. Series A 175 2020.10

Licci Level Stanley-Reisner Ideals with Height Three and with Type Two Invited Reviewed

Giancarlo Rinaldo, Naoki Terai, Ken Ichi Yoshida

Springer Proceedings in Mathematics and Statistics 331 123 - 142 2020

Authorship：Corresponding author Publishing type：Research paper (international conference proceedings)

© 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

Level property of ordinary and symbolic powers of Stanley-Reisner ideals Reviewed

Nguyên Công Minh, Naoki Terai, Phan Thi Thuy

Journal of Algebra 535 350 - 364 2019.10

Publishing type：Research paper (scientific journal)

© 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

4-Dimensional Licci Gorenstein Stanley-Reisner Ideals Invited Reviewed

Giancarlo Rinaldo, Naoki Terai

Acta Mathematica Vietnamica 44 ( 3 ) 691 - 700 2019.9

Depth and regularity modulo a principal ideal Reviewed

Giulio Caviglia, Huy Tài Hà, Jürgen Herzog, Manoj Kummini, Naoki Terai, Ngo Viet Trung

Journal of Algebraic Combinatorics 49 ( 1 ) 2019.2

Publishing type：Research paper (scientific journal)

© 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

Cohen-Macaulay and (S<inf>2</inf>) properties of the second power of squarefree monomial ideals Invited Reviewed

Do Trong Hoang, Giancarlo Rinaldo, Naoki Terai

Mathematics 7 ( 8 ) 2019

Authorship：Corresponding author Publishing type：Research paper (scientific journal)

© 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

The projective dimension of the edge ideal of a very well-covered graph Reviewed

Kyouko Kimura, Naoki Terai, Siamak Yassemi

Nagoya Mathematical Journal 230 160 - 179 2018.6

Publishing type：Research paper (scientific journal)

© 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

Stability of depths of symbolic powers of Stanley–Reisner ideals Reviewed

Le Tuan Hoa, Kyouko Kimura, Naoki Terai, Tran Nam Trung

Journal of Algebra 473 307 - 323 2017.3

Publishing type：Research paper (scientific journal)

© 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

Arithmetical Rank of a Squarefree Monomial Ideal whose Alexander Dual is of Deviation Two Reviewed

Kyouko Kimura, Naoki Terai, Ken ichi Yoshida

Acta Mathematica Vietnamica 40 ( 3 ) 375 - 391 2015.9

Publishing type：Research paper (scientific journal)

© 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

Multiplicity and Castelnuovo–Mumford Regularity of Stanley–Reisner Rings Invited Reviewed

Naoki Terai, Ken ichi Yoshida

Acta Mathematica Vietnamica 40 ( 1 ) 61 - 69 2015.3

Publishing type：Research paper (scientific journal)

© 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

Gorenstein and S<inf>r</inf> path ideals of cycles Reviewed

Dariush Kiani, Sara Seedi Madani, Naoki Terai

Glasgow Mathematical Journal 57 ( 1 ) 7 - 15 2015.1

Publishing type：Research paper (scientific journal)

© 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

Arithmetical rank of Gorenstein squarefree monomial ideals of height three Reviewed

Kyouko Kimura, Naoki Terai

Journal of Algebra 422 11 - 32 2015.1

Cohen-Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals Reviewed

A. Constantinescu, M. R. Pournaki, S. A. Seyed Fakhari, N. Terai, S. Yassemi

Communications in Algebra 43 ( 1 ) 143 - 157 2015.1

Publishing type：Research paper (scientific journal)

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

On the associated primes and the depth of the second power of squarefree monomial ideals Reviewed

Naoki Terai, Ngo Viet Trung

Journal of Pure and Applied Algebra 218 ( 6 ) 1117 - 1129 2014.6

Survey article: Simplicial complexes satisfying Serre's condition: A survey with some new results Invited Reviewed

M. R. Pournaki, S. A. Seyed Fakhari, N. Terai, S. Yassemi

Journal of Commutative Algebra 6 ( 4 ) 455 - 483 2014

Publishing type：Research paper (scientific journal)

© 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

Licci squarefree monomial ideals generated in degree two or with deviation two Reviewed

Kyouko Kimura, Naoki Terai, Ken ichi Yoshida

Journal of Algebra 390 264 - 289 2013.9

Publishing type：Research paper (scientific journal)

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

Binomial arithmetical rank of edge ideals of forests Reviewed

Kyouko Kimura, Naoki Terai

Proceedings of the American Mathematical Society 141 ( 6 ) 1925 - 1932 2013

Arithmetical Rank of Squarefree Monomial Ideals Generated by Five Elements or with Arithmetic Degree Four Reviewed

Kyouko Kimura, Giancarlo Rinaldo, Naoki Terai

Communications in Algebra 40 ( 11 ) 4147 - 4170 2012.11

Pure and Cohen-Macaulay Simplicial Complexes Associated with Squarefree Lexsegment Ideals Reviewed

Vittoria Bonanzinga, Loredana Sorrenti, Naoki Terai

Communications in Algebra 40 ( 11 ) 4195 - 4214 2012.11

Cohen-Macaulayness of large powers of Stanley-Reisner ideals Reviewed

Naoki Terai, Ngo Viet Trung

Advances in Mathematics 229 ( 2 ) 711 - 730 2012.1

Publishing type：Research paper (scientific journal)

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

Cohen-Macaulayness for symbolic power ideals of edge ideals Reviewed

Giancarlo Rinaldo, Naoki Terai, Ken ichi Yoshida

Journal of Algebra 347 ( 1 ) 1 - 22 2011.12

Publishing type：Research paper (scientific journal)

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

Vertex decomposability and regularity of very well-covered graphs Reviewed

Mohammad Mahmoudi, Amir Mousivand, Marilena Crupi, Giancarlo Rinaldo, Naoki Terai, Siamak Yassemi

Journal of Pure and Applied Algebra 215 ( 10 ) 2473 - 2480 2011.10

Publishing type：Research paper (scientific journal)

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

Schmitt-Vogel type lemma for reductions Reviewed

Kyouko Kimura, Naoki Terai, Ken ichi Yoshida

Archiv der Mathematik 96 ( 6 ) 535 - 545 2011.6

Sequentially S<inf>r</inf> simplicial complexes and sequentially S <inf>2</inf> graphs Reviewed

Hassan Haghighi, Naoki Terai, Siamak Yassemi, Rahim Zaare-Nahandi

Proceedings of the American Mathematical Society 139 ( 6 ) 1993 - 2005 2011.6

Publishing type：Research paper (scientific journal)

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

Cohen-macaulay edge ideal whose height is half of the number of vertices Reviewed

Marilena Crupi, Giancarlo Rinaldo, Naoki Terai

Nagoya Mathematical Journal 201 117 - 131 2011.3

The stanley-reisner ideals of polygons as set-theoretic complete intersections Reviewed

Margherita Barile, Naoki Terai

Communications in Algebra 39 ( 2 ) 621 - 633 2011.2

On the second powers of stanley-reisner ideals Reviewed

Giancarlo Rinaldo, Naoki Terai, Ken ichi Yoshida

Journal of Commutative Algebra 3 ( 3 ) 405 - 430 2011

Publishing type：Research paper (scientific journal)

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

Sequentially cohen-Macaulay path ideals of cycles

Sara Saeedi Madani, Dariush Kiani, Naoki Terai

Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie 54 ( 4 ) 353 - 363 2011

Publishing type：Research paper (scientific journal)

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.

Arithmetical ranks of stanley-reisner ideals of simplicial complexes with a cone Reviewed

Margherita Barile, Naoki Terai

Communications in Algebra 38 ( 10 ) 3686 - 3698 2010.10

Publishing type：Research paper (scientific journal)

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

Effective cowsik-nori theorem for edge ideals Reviewed

Marilena Crupi, Giancarlo Rinaldo, Naoki Terai, Ken ichi Yoshida

Communications in Algebra 38 ( 9 ) 3347 - 3357 2010

Arithmetical rank of lexsegment edge ideals Reviewed

Viviana Ene, Oana Olteanu, Naoki Terai

Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie 53 ( 4 ) 315 - 327 2010

H-vectors of simplicial complexes with Serre's conditions Reviewed

Satoshi Murai, Naoki Terai

Mathematical Research Letters 16 ( 6 ) 1015 - 1028 2009.11

Publishing type：Research paper (scientific journal)

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

Arithmetical rank of squarefree monomial ideals of small arithmetic degree Reviewed

Kyouko Kimura, Naoki Terai, Ken Ichi Yoshida

Journal of Algebraic Combinatorics 29 ( 3 ) 389 - 404 2009.5

Publishing type：Research paper (scientific journal)

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

Locally complete intersection stanley-reisner ideals Reviewed

Naoki Terai, Ken Ichi Yoshida

Illinois Journal of Mathematics 53 ( 2 ) 413 - 429 2009

Publishing type：Research paper (scientific journal)

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

A note on cohen-macaulayness of stanley-reisner rings with serre's condition (S2) Reviewed

Naoki Terai, Ken Ichi Yoshida

Communications in Algebra 36 ( 2 ) 464 - 477 2008.2

Publishing type：Research paper (scientific journal)

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

Stanley-Reisner rings with large multiplicities are Cohen-Macaulay Reviewed

Naoki Terai, Ken ichi Yoshida

Journal of Algebra 301 ( 2 ) 493 - 508 2006.7

Buchsbaum stanley-reisner rings and cohen-macaulay covers Reviewed

Naoki Terai, Ken Ichi Yoshida

Communications in Algebra 34 ( 7 ) 2673 - 2681 2006.6

Publishing type：Research paper (scientific journal)

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

Buchsbaum Stanley-Reisner rings with minimal multiplicity Reviewed

Naoki Terai, Ken Ichi Yoshida

Proceedings of the American Mathematical Society 134 ( 1 ) 55 - 65 2006.1

Publishing type：Research paper (scientific journal)

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

On the radical of a monomial ideal Reviewed

J. Herzog, Y. Takayama, N. Terai

Archiv der Mathematik 85 ( 5 ) 397 - 408 2005.11

Castelnuovo--Mumford regularity and initial ideals with no embedded prime ideal Reviewed

Naoki Terai, Hidefumi Ohsugi, Takayuki Hibi

Acta Math. Vietnam 29 135 - 139 2004

Stable Properties of Algebraic Shifting Reviewed

JÜrgen Herzog, Naoki Terai

Results in Mathematics 35 ( 3-4 ) 260 - 265 1999.5

Computation of Betti numbers of monomial ideals associated with stacked polytopes Reviewed

Naoki Terai, Takayuki Hibi

Manuscripta Mathematica 92 ( 4 ) 447 - 453 1997.4

Publishing type：Research paper (scientific journal)

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

Finite Free Resolutions and 1-Skeletons of Simplicial Complexes Reviewed

Naoki Terai, Takayuki Hibi

Journal of Algebraic Combinatorics 6 ( 1 ) 89 - 93 1997

Alexander duality theorem and second betti numbers of Stanley-Reisner rings Reviewed

Naoki Terai, Takayuki Hibi

Advances in Mathematics 124 ( 2 ) 332 - 333 1996.12

Some results on Betti numbers of Stanley-Reisner rings Reviewed

Naoki Terai, Takayuki Hibi

Discrete Mathematics 157 ( 1-3 ) 311 - 320 1996.10

Publishing type：Research paper (scientific journal)

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

Computation of Betti numbers of monomial ideals associated with cyclic polytopes Reviewed

N. Terai, T. Hibi

Discrete and Computational Geometry 15 ( 3 ) 287 - 295 1996.4

Publishing type：Research paper (scientific journal)

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

On h-vectors of Buchsbaum Stanley-Reisner rings Reviewed

Naoki Terai

Hokkaido Mathematical Journal 25 ( 1 ) 137 - 148 1996

Publishing type：Research paper (scientific journal)

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

Some remarks on algebras with straightening laws Reviewed

Naoki Terai

Journal of Pure and Applied Algebra 95 ( 1 ) 87 - 101 1994.7

Authorship：Lead author Publishing type：Research paper (scientific journal)

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

Homogeneous complete intersection Hodge algebras on simplicial complexes Reviewed

Naoki Terai

Osaka Journal of Mathematics 31 ( 2 ) 341 - 353 1994.6

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