Faculty Profiles - TERAI Naoki

Papers - TERAI Naoki

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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

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