Organization
Faculty of Engineering Science Professor
Title
Professor
Contact information
External link
YANAGAWA,Kohji
Degree
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Doctor of Science ( 1996.3 )
Research Interests
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combinatorics
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commutative algebra
Research Areas
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Natural Science / Algebra
Education
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Nagoya University Graduate School, Division of Natural Science
- 1996
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Nagoya University Faculty of Science
- 1991
Country： Japan
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Nagoya University Graduate School, Division of Natural Science
1996
Country： Japan
Research History
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1996-1997 Niigata University, Research Assistant 1997-2007 Osaka University, Research Assistant2007- Kansai University Associate Professor
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1996-1997 Niigata University, Research Assistant1997-2007 Osaka University, Research Assistant2007- Kansai University Associate Professor
Professional Memberships
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Mathematical Society of Japan
Papers
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Arithmetic on $q$-deformed rational numbers Reviewed
Takeyoshi Kogiso, Kengo Miyamoto, Xin Ren, Michihisa Wakui, Kohji Yanagawa
Arnold Mathematical Journal 011 ( 003 ) 42 - 92 2025.10
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Elementary construction of the minimal free resolution of the Specht ideal of shape (n − d,d) Reviewed
Kosuke Shibata, Kohji Yanagawa
Journal of Algebra Vol 634, no. 15, pp. 563-584 563 - 584 2023.11
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Gröbner Bases of Radical Li–Li Type Ideals Associated with Partitions Reviewed
Xin Ren, Kohji Yanagawa
SIAM Journal on Discrete Mathematics Vol. 37, No. 4, ( 4 ) 2382 - 2396 2023.10
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Elementary construction of minimal free resolutions of the Specht ideals of shapes (n−2,2) and (d,d,1) Reviewed
Shibata, Kosuke, Yanagawa, Kohji
Journal of Algebra and Its Applications Vol. 22, No. 9 2350199 2023.9
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A note on the reducedness and Grobner bases of Specht ideals Reviewed
Satoshi Murai, Hidefumi Ohsugi, Kohji Yanagawa
COMMUNICATIONS IN ALGEBRA Vol. 50,pp. 5430-5434 ( 12 ) 5430 - 5434 2022.12
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Graded Cohen–Macaulay Domains and Lattice Polytopes with Short h-Vector Reviewed
Katthän, Lukas, Yanagawa, Kohji
Discrete & Computational Geometry Volume 68, issue 2, pp. 608–617 2022.9
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Regularity of Cohen-Macaulay Specht ideals Reviewed
SHIBATA, Kosuke, YANAGAWA, Kohji
Journal of Algebra Vol. 582, Pages 73-87 73 - 87 2021.9
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When is a Specht ideal Cohen–Macaulay? Reviewed
YANAGAWA,Kohji
Journal of Commutative Algebra Volume 13 , No. 4, 589–608 2021
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Alexander duality for the alternative polarizations of strongly stable ideals Reviewed
SHIBATA, Kosuke, YANAGAWA,Kohji
Communications in Algebra Vol. 48, pp. 3011-3030 2020.7
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Vandermonde determinantal ideals Reviewed
WATANABE, Junzo, YANAGAWA, Kohji
Mathematica Scandinavica Vol 125, No 2, pp.179-184 2019.10
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LYUBEZNIK NUMBERS OF LOCAL RINGS AND LINEAR STRANDS OF GRADED IDEALS Reviewed
ÀLVAREZ MONTANER, Josep, YANAGAWA,Kohji
Nagoya Mathematical Journal Volume 231,pp. 23-54 2018.9
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The Cohen–Macaulayness of the bounded complex of an affine oriented matroid Reviewed
Ryota Okazaki, Kohji Yanagawa
Journal of Combinatorial Theory. Series A 157 1 - 27 2018.7
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Non-level semi-standard graded Cohen-Macaulay domain with h-vector (h(0), h(1), h(2)) Reviewed
Akihiro Higashitani, Kohji Yanagawa
JOURNAL OF PURE AND APPLIED ALGEBRA 222 ( 1 ) 191 - 201 2018.1
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A canonical module characterization of Serre's (R-1) Reviewed
Lukas Katthaen, Kohji Yanagawa
COMMUNICATIONS IN ALGEBRA 45 ( 2 ) 600 - 605 2017
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Properties of Lyubeznik numbers under localization and polarization Reviewed
Arindam Banerjee, Luis Nunez-Betancourt, Kohji Yanagawa
JOURNAL OF PURE AND APPLIED ALGEBRA 219 ( 11 ) 4872 - 4888 2015.11
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Dualizing complexes of seminormal affine semigroup rings and toric face rings Reviewed
Kohji Yanagawa
JOURNAL OF ALGEBRA 425 367 - 391 2015.3
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On CW complexes supporting Eliahou-Kervaire type resolutions of Borel fixed ideals Reviewed
Ryota Okazaki, Kohji Yanagawa
COLLECTANEA MATHEMATICA 66 ( 1 ) 125 - 147 2015.1
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Squarefree P-modules and the cd-index Reviewed
Satoshi Murai, Kohji Yanagawa
ADVANCES IN MATHEMATICS 265 241 - 279 2014.11
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Addendum to "Frobenius and Cartier algebras of Stanley-Reisner rings" [J. Algebra 358 (2012) 162-177] Reviewed
Josep Àlvarez Montaner, Kohji Yanagawa
Journal of Algebra 414 300 - 304 2014.9
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Alternative polarizations of Borel fixed ideals, Eliahou-Kervaire type resolution and discrete Morse theory Reviewed
Ryota Okazaki, Kohji Yanagawa
JOURNAL OF ALGEBRAIC COMBINATORICS 38 ( 2 ) 407 - 436 2013.9
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ALTERNATIVE POLARIZATIONS OF BOREL FIXED IDEALS Reviewed
Kohji Yanagawa
NAGOYA MATHEMATICAL JOURNAL 207 79 - 93 2012.9
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SLIDING FUNCTOR AND POLARIZATION FUNCTOR FOR MULTIGRADED MODULES Reviewed
Kohji Yanagawa
COMMUNICATIONS IN ALGEBRA 40 ( 3 ) 1151 - 1166 2012
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HIGHER COHEN-MACAULAY PROPERTY OF SQUAREFREE MODULES AND SIMPLICIAL POSETS Reviewed
Kohji Yanagawa
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 139 ( 9 ) 3057 - 3066 2011.9
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Dualizing complex of the face ring of a simplicial poset Reviewed
Kohji Yanagawa
JOURNAL OF PURE AND APPLIED ALGEBRA 215 ( 9 ) 2231 - 2241 2011.9
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Alexander duality and Stanley depth of multigraded modules Reviewed
Ryota Okazaki, Kohji Yanagawa
JOURNAL OF ALGEBRA 340 ( 1 ) 35 - 52 2011.8
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DUALIZING COMPLEX OF A TORIC FACE RING Reviewed
Ryota Okazaki, Kohji Yanagawa
NAGOYA MATHEMATICAL JOURNAL 196 87 - 116 2009.12
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LINEARITY DEFECT AND REGULARITY OVER A KOSZUL ALGEBRA Reviewed
Kohji Yanagawa
MATHEMATICA SCANDINAVICA 104 ( 2 ) 205 - 220 2009
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Notes on C-graded modules over an affine semigroup ring K[C] Reviewed
Kohji Yanagawa
COMMUNICATIONS IN ALGEBRA 36 ( 8 ) 3122 - 3146 2008
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Linearity defects of face rings Reviewed
Ryota Okazaki, Kohji Yanagawa
JOURNAL OF ALGEBRA 314 ( 1 ) 362 - 382 2007.8
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BGG correspondence and Romer's theorem on an exterior algebra Reviewed
Kohji Yanagawa
ALGEBRAS AND REPRESENTATION THEORY 9 ( 6 ) 569 - 579 2006.12
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Castelnuovo-Mumford regularity for complexes and weakly Koszul modules Reviewed
Kohji Yanagawa
JOURNAL OF PURE AND APPLIED ALGEBRA 207 ( 1 ) 77 - 97 2006.9
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Dualizing complex of the incidence algebra of a finite regular cell complex Reviewed
YANAGAWA Kohji
Illinois Journal of Mathematics Vol.49, no.4, pp.1221--1243. 2005
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Derived category of squarefree modules and local cohomology with monomial ideal support Reviewed
K Yanagawa
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 56 ( 1 ) 289 - 308 2004.1
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Stanley-Reisner rings, sheaves, and Poincart-Verdier duality Reviewed
K Yanagawa
MATHEMATICAL RESEARCH LETTERS 10 ( 5-6 ) 635 - 650 2003.9
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Local cohomology of Stanley-Reisner rings with supports in general monomial ideals Reviewed
Reiner, V, Welker, V, K Yanagawa
JOURNAL OF ALGEBRA 244 ( 2 ) 706 - 736 2001.10
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Bass numbers of local cohomology modules with supports in monomial ideals Reviewed
K Yanagawa
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY 131 45 - 60 2001.7
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Sheaves on finite posets and modules over normal semigroup rings Reviewed
K Yanagawa
JOURNAL OF PURE AND APPLIED ALGEBRA 161 ( 3 ) 341 - 366 2001.7
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Generic and cogeneric monomial ideals Reviewed
E Miller, B Sturmfels, K Yanagawa
JOURNAL OF SYMBOLIC COMPUTATION 29 ( 4-5 ) 691 - 708 2000.4
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Alexander duality for Stanley-Reisner rings and squarefree N-n-graded modules Reviewed
K Yanagawa
JOURNAL OF ALGEBRA 225 ( 2 ) 630 - 645 2000.3
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F-Delta type free resolutions of monomial ideals Reviewed
K Yanagawa
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 127 ( 2 ) 377 - 383 1999.2
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Squarefree modules and local cohomology modules at monomial ideals Invited Reviewed
K Yanagawa
LOCAL COHOMOLOGY AND ITS APPLICATIONS 226 207 - 231 1999
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Associated primes and arithmetic degrees Reviewed
C Miyazaki, W Vogel, K Yanagawa
JOURNAL OF ALGEBRA 192 ( 1 ) 166 - 182 1997.6
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On the regularities of arithmetically Buchsbaum curves Reviewed
Kohji Yanagawa
Mathematische Zeitschrift 226 ( 1 ) 155 - 163 1997
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A characterization of integral curves with Gorenstein hyperplane sections Reviewed
K Yanagawa
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 124 ( 5 ) 1379 - 1384 1996.5
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Castelnuovo's lemma and h-vectors of Cohen-Macaulay homogeneous domains Reviewed
K Yanagawa
JOURNAL OF PURE AND APPLIED ALGEBRA 105 ( 1 ) 107 - 116 1995.11
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SOME GENERALIZATIONS OF CASTELNUOVO LEMMA ON ZERO-DIMENSIONAL SCHEMES Reviewed
K YANAGAWA
JOURNAL OF ALGEBRA 170 ( 2 ) 429 - 439 1994.12
Presentations
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Transposes in the q-deformed modular group and their applications to q-deformed rational numbers
任 鑫, 柳川 浩二
日本数学会2025年度秋季総合分科会 2025.9
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単体的ポセットの組合せ論的可換代数 Invited
柳川 浩二
第29回代数学若手研究会 2025.3
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Arithmetic and combinatorics on q-deformed rationals Invited
Commutative Algebra and Projective Algebraic Geometry in Kumamoto, -- on the occasion of Prof. Miyazaki's 65 2024.12
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Local cohomology modules at ideals associated with simplicial posets
Kohji Yanagawa
RIMS Workshop "The 45th Japan Symposium on Commutative Algebra" 2024.11
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Lyubeznik numbers of Stanley-Reisner ideals, and more... Invited
柳川浩二
空間の代数的・幾何的モデルとその周辺 2024.8
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Gr¨obner bases of radical Li–Li type ideals associated with partitions
MSJ Autumn Meeting 2023 2023.9
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Regularity of Cohen-Macaulay Specht ideals
柴田 孝祐, 柳川 浩二
第41回可換環論シンポジウム 2019.11
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Squarefree 加群とその応用, I, II
柳川 浩二
組合せ論と可換代数オータムセミナー 2019.9
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Specht ideal による剰余環の Cohen-Macaulay 性 Invited
柳川 浩二
東京可換環論セミナー 2019.4
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When is a Specht ideal Cohen-Macaulay?
YANAGAWA,Kohji
1147th AMS Meeting, Spring Central and Western Joint Sectional Meeting, Special Session on Commutative Algebra and its Environs, 2019.3
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Strongly stable ideal の既約分解と局所コホモロジーの関係
柴田 孝祐, 柳川 浩二
日本数学会 2019年度年会 代数学分科会 2019.3
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Strongly stable ideal のalternative polarization とそのAlexander 双対 について
柴田 孝祐, 柳川 浩二
日本数学会 秋季総合分科会 代数学分科会 2018.9
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Homological methods in combinatorial commutative algebra I, II
YANAGAWA,Kohji
The 51st Symposium on Ring Theory and Representation Theory 2018.9
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Stanley-Reisner 環の理論におけるホモロジカルな手法, I,II
柳川 浩二
第８回 (非)可換代数とトポロジー 2018.2
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When is a Specht ideal Cohen-Macaulay?
YANAGAWA,Kohji
The 39th Japan Symposium on Commutative Algebra 2017.11
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When is a Specht ideal Cohen-Macaulay?
YANAGAWA,Kohji
2017.8
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Non-level semi-standard graded Cohen-Macaulay domains with h-vectors (h_0, h_1, h_2)
Akihiro Higashitani, Kohji Yanagawa
The 38th Symposium on Commutative Algebra 2016.11
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Non-level semi-standard graded Cohen-Macaulay domains with h-vectors (h_0, h_1, h_2)
東谷 章弘, 柳川 浩二
日本数学会 秋季総合分科会 代数学分科会 2016.9
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Lyubeznik numbers of local rings and linear strands of graded ideals
YANAGAWA,Kohji
2016.3
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The Cohen-Macaulayness of the bounded complex of an affine oriented matroid
OKAZAKI, Ryota, YANAGAWA,Kohji
2015.11
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アフィン有向マトロイドのbounded complex のCohen-Macaulay 性とマトロイド・イデアルのCohen-Macaulay性
岡崎 亮太, 柳川 浩二
日本数学会 秋季総合分科会 代数学分科会 2015.9
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The Cohen-Macaulayness of the bounded complex of an affine oriented
YANAGAWA,Kohji
2015.8
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Serre の(R1) 条件の標準加群による特徴づけ
柳川 浩二
第7回岡山可換代数表現セミナー 2015.6
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Lyubeznik numbers of local rings and linear strands of graded ideals
Kohji Yanagawa
The 36th Symposium on Commutative Algebra 2014.11
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単項式イデアルの Lyubeznik 数 ～ polarization と局所化
柳川 浩二
2014.11
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単項式イデアルの Lyubeznik 数 ～ polarizationと局所化
柳川 浩二
第4回岡山可換代数表現セミナー 2014.11
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有向マトロイドに付随する単項式イデアル
柳川 浩二
組合せ論と可換代数サマーセミナー２０１４ 2014.9
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多面体的複体の flag f-列について
村井聡, 柳川浩二
日本数学会2014年度年会 代数学分科会 2014.3
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単項式イデアルの Lyubeznik table
柳川 浩二
組合せ論と可換代数サマーセミナー 2013.8
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Dualizing complexes of seminormal affine semigroup rings and toric face rings
Kohji Yanagawa
The 34th symposium on commutative algebra in Japan 2012.11
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On a minimal free resolution of a Borel fixed ideal and its supporting CW complex
Ryota Okazaki, Kohji Yanagawa
The Mathematical Society of Japan, Autumn Meeting 2012 2012.9
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Free resolutions of (variants of) Borel fixed ideals
Kohji Yanagawa
The 52nd Algebra Symposium 2012.8
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Alternative polarizations of Borel fixed ideals, Eliahou-Kervaire type resolution and discrete Morse theory
Yanagawa, Kohji
The 7th Japan-Vietnam Joint Seminar on Commutative Algebra 2011.12
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Alternative polarizations of Borel fixed ideals and Eliahou-Kervaire type
Ryota Okazaki, Kohji Yanagawa
2011.9
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Derived categories and topological methods in combinatorial commutative algebra I, II
YANAGAWA Kohji
2011.9
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Sliding functor and polarization functor for multigraded modules
YANAGAWA Kohji
The 6th Japan-Vietnam Joint Seminar on Commutative Algebra 2010.12
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On the face ring of a simplicial poset
YANAGAWA Kohji
The 5-th Japan-Vietnam Joint Seminar on Commutative Algebra 2010.1
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Recent topics on derived categories in combinatorial commutative algebra
YANAGAWA Kohji
Workshop on "Syzygies of Projective Varieties" 2009.9
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Dualizing complex of a toric face ring
YANAGAWA Kohji
Pacific Rim Mathematical Association (PRIMA) Congress 2009.6
Works
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MFO-RIMS Tandem Workshop: Symmetries on Polynomial Ideals and Varieties (hybrid meeting)
柳川 浩二
2021.9
Research Projects
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組合せ論的可換代数への圏論的アプローチの深化
Grant number：25K06928 2025.4 - 2029.3
日本学術振興会 科学研究費助成事業 基盤研究(C)
柳川 浩二
Grant amount：\3770000 （ Direct Cost: \2900000 、 Indirect Cost：\870000 ）
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The study on ring theoretic properties and Groebner basis of Specht ideals
Grant number：22K03258 2022.4 - 2025.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
Grant amount：\2210000 （ Direct Cost: \1700000 、 Indirect Cost：\510000 ）
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The Cohen-Macaulay property of ideals associated with subspace arrangements
Grant number：19K03456 2019.4 - 2022.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
Grant amount：\2990000 （ Direct Cost: \2300000 、 Indirect Cost：\690000 ）
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Application of commutative algebra to topological study on affine oriented matroids
Grant number：16K05114 2016.4 - 2020.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
Yanagawa Kohji
Grant amount：\3250000 （ Direct Cost: \2500000 、 Indirect Cost：\750000 ）
Just before this project started, the author and his coworker (almost ) showed that if the ideal associated with an affine oriented matroid M is Cohen-Macaulay (CM,for short), then the bounded complex of M is a contractible homology manifold (with boundary). In this situation, we conjectured that the bounded complex is homeomorphic to a closed ball. This conjecture is the main aim of this project. Finally, we proved the conjecture when the dimension is at most 3. We also showed that the bounded complex is a topological manifold in the dimension 4 case.
In the latter period of this project, the Specht ideals have been a main object of the study.We completely determined the CM Specht ideals in the characteristic 0 case. -
Arithmetical rank of Stanley-Reisner ideals and projective dimension of their powers
Grant number：26400049 2014.4 - 2017.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Terai Naoki, YOSHIDA KENICHI, YANAGAWA KOUJI, KIMURA KYOUKO
Grant amount：\4810000 （ Direct Cost: \3700000 、 Indirect Cost：\1110000 ）
We studied the projective dimension of symbolic powers of squarefree monomial ideals in a polynomial ring. We proved that the projective dimension of the symbolic power of the edge ideal of a very well-covered graph increases with respect to the exponent. Since a well-covered bipartite graph is very well-covered and since the symbolic powers and ordinary powers coincide for the edge ideal of a bipartite graph, it implies that the projective dimension of the ordinary power of the edge ideal of a very well-covered graph increases. Moreover, we showed that the projective dimension of the symbolic power of the edge ideal of a graph with a vertex of degree one increases with respect to the exponent.
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Application of New Methods of Combinatorial Topology to Commutative Algebra
Grant number：25400057 2013.4 - 2016.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
Yanagawa Kohji
Grant amount：\2860000 （ Direct Cost: \2200000 、 Indirect Cost：\660000 ）
With Spanish and American coauthors, I studied the Lyubeznik numbers of the quotient rings of polynomial rings by monomial ideals. For example, we showed that the procedure of "polarization" essentially preserves this invariant. With Prof. S. Murai, I studied the "cd-index" of flag complexes using the notion of squarefree modules. I also studied cellular free resolutions of monomial ideals with Prof. R. Okazaki. We applied this method to the study of affine oriented matroids.
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Minimal free resolutions and the arithmetical rank of Stanley-Reisner ideals
Grant number：23540053 2011 - 2013
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
TERAI Naoki, UEHARA Tsuyoshi, ICHIKAWA Takashi, MIYAZAKI Chikashi, KAWAI Shigeo, YOSHIDA Kenichi, YANAGAWA Kouji, KIMURA Kyouko, MURAI Satoshi
Grant amount：\5070000 （ Direct Cost: \3900000 、 Indirect Cost：\1170000 ）
We studied the arithmetical rank of Stanly-Reisner ideals, which are squarefree monomial ideals in a polynomial ring. It is known that the arithmetical rank of a Stanley-Reisner ideal is greater than or equal to the projective dimension of the Stanley-Reisner ring, which is the length of the minimal free resolutions of the quotient ring. As for the edge ideal of a forest, Barile conjectures that these numbers will be coincident. We proved it. As for a Gorenstein Stanly-Reisner ideal of height three, we proved that its arithmetical rank is equal to the projective dimension of the Stanly-Reisner ring, too.
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The applications of derived categories and topological methods to combinatorial commutative algebra
Grant number：22540057 2010 - 2012
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
YANAGAWA Kohji
Grant amount：\2990000 （ Direct Cost: \2300000 、 Indirect Cost：\690000 ）
Developing a result of Nagel and Reiner, the author and R. Okazaki studied the non-standard polarization of a Borel fixed ideal, and gave its minimal free resolution. This resolution is supported by a regular CW complex, which can be “interpreted” by the discrete Morse theory in the framework constructed by Welker et al.
The author also studied the dualizing complexes of seminormal affine semigroup rings and toric face rings. In this work, the preceding results of Bruns et al and Nguyes are re-formulated and improved using derived categories. -
Application of Koszul duality to commutative algebra
Grant number：19540028 2007 - 2009
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C) Grant-in-Aid for Scientific Research (C)
YANAGAWA Kouji, TERAI Naoki, WAKUI Michihisa, MORI Izuru, WAKAMATSU Takayoshi
Grant amount：\3900000 （ Direct Cost: \3000000 、 Indirect Cost：\900000 ）
The author has been applied the theory of derived categories (e.g., Koszul duality) to the study of combinatorial commutative algebra. The main aim of this research project was to extend this idea and results to general commutative algebra. In this direction, results related to the coherent property of rings were obtained, and the paper was published in an academic journal in 2009. Around 2008, the author slightly changed the direction of the research, and begun to study relatively new objects of combinatorial commutative algebra for which we cannot use usual methods. In this direction, the author has written several papers. Some of them have been published in an academic journal.
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THE RING OF DIFFERENTIAL OPERATORS ON AN AFFINE TORIC VARIETYAND ITS APPLICATIONS
Grant number：18540002 2006 - 2008
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
SAITO Mutsumi, YAMASHITA Hirosh, YANAGAWA Kohji, SHIMADA Ichiro, NUMATA Yasuhide, YANAGAWA Kohji, SHIMADA Ichiro, NUMATA Yasuhide
Grant amount：\4060000 （ Direct Cost: \3400000 、 Indirect Cost：\660000 ）
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Study on minimal free resolution of Stanley-Reisner rings
Grant number：18540041 2006 - 2007
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
TERAI Naoki, NAKAHARA Tohru, UEHARA Tsuyoshi, ICHIKAWA Takashi, YOSHIDA Ken-ichi, YANAGAWA Kohji
Grant amount：\4010000 （ Direct Cost: \3500000 、 Indirect Cost：\510000 ）
The purpose of this research is to study algebraic and combinatorial properties of minimal free resolution of Stanley-Reisner rings. We focused on the relation between the multiplicity and the Castelnuovo-Mumford regularity of Stanley-Reisner rings.
Before the academic year 2005 we proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to the dimension d of the Stanley-Reisner rings if its multiplicity is less than or equal to d. Moreover we verified that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 2d-1, and if the degree of generators of the Stanley-Reisner ideal is less than or equal to d
In the academic year 2006, developing these results, we proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 3d-2, and if the degree of generators of the first syzygy module of the Stanley-Reisner ideal is less than or equal to d+1. From this result we conjectured that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to (p+2)d- (p-1), and if the degree of generators of the p-th syzygy module of the Stanley-Reisner ideal is less than or equal to d+p. In the academic year 2007 we proved that the above conjecture holds if the dimension of the Stanley-Reisner rings is 2 or 3. We also found that this conjecture is a generalization of the lower bound theorem, that is famous in convex polytope theory, in the facet case. -
Study on minimal free resolution of Stanley-Reisner rings
Grant number：16540028 2004 - 2005
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
TERAI Naoki, TANAKA Tatsuji, NAKAHARA Tohru, ICHIKAWA Takashi, YOSHIDA Ken-ichi, YANAGAWA Kohji
Grant amount：\3600000 （ Direct Cost: \3600000 ）
The purpose of this research is to study algebraic and combinatorial properties of minimal free resolution of Stanley-Reisner rings and to consider its combinatorial applications.
In the academic year 2004 we studied Buchsbaum Stanley-Reisner rings with linear resolution. We determined the lower bound for the multiplicity of Stanley-Reisner rings. And we showed that they have linear resolution if they possess the minimal multiplicity. We also showed a necessary and sufficient condition for Buchsbaum Stanley-Reisner rings to have linear resolution in terms of the reduced homology groups of the corresponding simplicial complex and its links.
In the academic year 2005 we mainly studied the relation between the multiplicity of Stanley-Reisner rings and their Castelnuovo-Mumford regularity. We proved that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to the dimension d if its multiplicity is less than or equal to d. Moreover we verified that the Castelnuovo-Mumford regularity of a Stanley-Reisner ideal is less than or equal to d if its multiplicity is less than or equal to 2d-1, and if the degree of generators of the Stanley-Reisner ideal is less than or equal to d
We also investigated Stanley-Reisner rings with d-linear resolution among those with linear resolution intensively. Using the above result we showed that a Stanley-Reisner ring has d-linear resolution if its multiplicity is less than or equal to d and if the degree of generators of the Stanley-Reisner ideal is more than or equal to d. Moreover we showed that a Stanley-Reisner ring has d-linear resolution if its multiplicity is less than or equal to 2d-1 and if the degree of generators of the Stanley-Reisner ideal is d. By Alexander duality, we also verified that a Stanley-Reisner ring is Cohen-Macaulay if its multiplicity is large enough. -
層や導来圏の理論を用いた組合せ論的可換代数の研究
Grant number：15740014 2003 - 2005
日本学術振興会 科学研究費助成事業 若手研究(B) 若手研究(B)
柳川 浩二
Grant amount：\3400000 （ Direct Cost: \3400000 ）
昨年度に執筆した論文"Dualizing complex of the incidence algebra of finite regular cell complex"と"Castelnuovo-Mumford regularity for complexes and weakly Koszul modules"が,今年度に入って学術雑誌に受理された.特に前者は,受理される前に大幅な加筆を行い,組合せ論的な側面(正則胞体分割の半順序集合としての「メビウス関数」など)を強化している.
また,今年度に得られた結果として,「アファイン半群環K[C]の被約な単項式イデアルIによる剰余環(最近では,"toric face ring"等と呼ばれるもの)の"sequentially Cohen-Macaulay"性(以下"seq.CM"と略す)は,Iに付随するpolytopal complexの位相的性質(と体Kの標数)のみによって定まる」を示したこと等が挙げられる.論文は,現在執筆中である.toric face ringのCohen-Macaulay性が位相的性質であることは(K[C]が多項式環の場合のMunkresの著名な結果の一般化であるが),20年前Stanleyによって証明されており,筆者も数年前,層の理論を用いた別証明を与えている.今回の結果は,この筆者自身の論法の発展である(Stanleyの証明は,Yuzvinskyによる"section ring"の理論の応用であるが,seq.CM性の場合に,この論法を用いるのは困難かと思われる).なお,seq.CMは,Cohen-Macaulayを一般化した概念で,近年では,"shifting"や"non-pure shellability"との関連から,「組合せ論的可換代数」での重要性が増している. -
Stanley-Reisner環の理論に現れるKoszul双対性の研究
Grant number：13740011 2001 - 2002
日本学術振興会 科学研究費助成事業 若手研究(B) 若手研究(B)
柳川 浩二
Grant amount：\1800000 （ Direct Cost: \1800000 ）
私は,数年前,組合せ論的可換代数の基本的な概念である「Stanley-Reisenr環」を一般化して,「squarefree加群」を定義した.この概念は,当該分野の道具の一つとして定着しつつある.squarefree加群の有界な導来圏には,3つの「双対性」が存在し,3次対称群の3つの互換の如く作用している.この現象は,Koszul双対性(特に,Bernstein-Gel'fand-Gel'fand対応)と関連している.この研究成果は,Journal of the Mathematical Society of Japanへの掲載が決定している"Derived Category of Squarefree Modules and Local Cohomology with Monomial Ideal Support"にまとめた.(この論文の第一稿は,平成12年度に投稿したが,レフェリーの助言もあり,平成13年度に全面的に改定し,結果を大幅に強めた.また,14年度にも若干の改良を行った.この意味で,この論文は13-14年度の成果でもある.)
Stanley-Reisener環が,対応する単体的複体の位相幾何学的性質を様々に反映しているのに対し,その一般化であるsquarefree加群の幾何学的な意味は,暫くはっきりしなかったが,平成14年度の研究で,(n変数多項式環上の)squarefree加群から,(n-1)次元閉球体上の層が構成でき,自然な(スキーム論における"Proj"と類似の)理論が展開できることが分かった.この文脈では,位相幾何学のポアンカレ・ヴェルディエ双対性と,可換代数の局所双対性が等価となる.この研究成果は,投稿中のプレプリント"Stanley-Reisner rings, sheaves, and Poincare-Verdier duality"にまとめた. -
単項式イデアルの研究への圏論的手法の応用
Grant number：12740013 2000
日本学術振興会 科学研究費助成事業 奨励研究(A)
柳川 浩二
Grant amount：\1100000 （ Direct Cost: \1100000 ）
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Geometry of space of Riemannian manifolds
Grant number：11640075 1999 - 2002
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
OTSU Yukio, GOTO Ryushi, SHIOYA Takashi, YAMADA Koutarou
Grant amount：\3600000 （ Direct Cost: \3600000 ）
Let us denote by A the space of Alexnadrov spaces of bounded curvature below and Hausdorff dimension above equipped with Hausdorff distance and by I the space of upper-semicontinuous functions on A. We call I the space of invariants. An ordered finite set of points of metric space is called a net, which is a discretization of the metric space. Since the configuration space of all nets is identified with the product of the space, the set N of pairs of spaces in A and its nets can be interpreted as a fiber space over A. We consider a map that assign the matrix of mutual distances of two points for each net. In this way we can represent N as a subspace of some Banach space. Then we introduce other maps form N to some Euclidian space that take local information of the above distance matrix. Especially we defined discrete Laplacian similar to the Laplacian of functions of Riemannian manifold. We introduced new statistical method to take average of discrete Laplacian on configuration space of nets. In this way we have showd that the eigenvalues and eigenvectors of discrete Laplacian converge to the limit independent of the choice of nets ; we also proved that coincides with the Laplacian in the sense of Kuwae-Machigashira-shioya in some sense.
Next we defined new structure on A by comparing two discrete Laplacian of different spaces and nets because they are same member of matrix space. Since in information geometry the relative entropy of two distributions determines Reimannian metric, we first introduced stationary Markov chain form the Laplacian, then we apply the relative entropy for them; finally we construct continuum limit of them, which is a generalization of Hausdorff distance. -
次数付可換環のヒルベルト関数と極小自由分解
Grant number：09740014 1997 - 1998
日本学術振興会 科学研究費助成事業 奨励研究(A) 奨励研究(A)
柳川 浩二
Grant amount：\1800000 （ Direct Cost: \1800000 ）
97年度は、E.Ballico 氏と共同で、very strange curve と言う、ある意味で極めて病的な射影曲線を研究し、これを正標数の代数閉体上の Cohen-Macaulay 斉次整域のヒルベルト関数の研究に応用した。very strange curve の定義イデアルは基礎体の標数より小さい次数の元を殆ど含まないことを示したものである(単なる“strangecurve"では、この性質を持たない)が、未だはっきりしない部分も多く、今後も研究を継続していきたい。
また同年度は、Bayer, Peeva, Sturmfels らによって当時導入されたばかりであった、凸幾何学的に特殊な構造を持った monomial ideal を研究し、これの準素成分がある種の「連結性」を持つことを示した(少し後で、toric ideal の initial ideal の準素成分が同様の性質を持つことが、 Hosten と Thomas によって示されている)。先行する研究が凸幾何学的手法によるものであったのに対し、筆者は、局所双対性等、可換代数的な手法も併用した。この研究は、98年度には、E.Mill,Sturmfels 両氏との共同研究に発展した(現在投稿中)。ここでは、 toric ideal の initial ideal も研究しており、手法的にも、 local h-vector や Alexander 双対性といった新しいものを導入している。
また、98年度は、 Stanley-Reisner 環の Alezander 双対性も研究し、 Cohen-Macaulayな Stanley-Reisner 環 k[Δ] の標準加群の加群構造の情報と、その双対の Stanley-Reisner 環 k[Δ^V]の極小自由分解の(微分写像まで込めた)情報が等価であること等を示した。
上述の monomial ideal や toric ideal の凸幾何学的自由分解の話題は、最近、Bayer,Popescu らによって(凸幾何学的にも可換代数的にも)より精密な方向に発展してきており、筆者も、この方向での研究を続けていきたいと考えている。 -
Foundation of computational Commutative algebra with a view toward combinatorics on convex polytopes
Grant number：09440013 1997 - 1998
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
HIBI Takayuki, YANAGAWA Koji, NAMIKAWA Yoshinori, MIYANISHI Masayoshi, SUZUKI Takashi, KAWANAKA Noriaki
Grant amount：\3400000 （ Direct Cost: \3400000 ）
The important activity during the period of the present research project is, first, to present the concept of componentwise linear ideals and to establish its fundamental theory and, second, to study generic initial ideals of simplicial complexes and to discuss their concrete and effective applications to combinatorics. First of all, we obtained the theorem that the squarefree monomial ideal associated with a simplicial complex is componentwise linear if and only if its dual complex is sequentially Cohen-Macaulay, and explained the algebraic aspect of sequentially Cohen-Macaulay complexes and their h-triangles. Second, based on fundamental study about generic initial ideals of coruponentwise linear ideals, the important result that a homogeneous ideal of the polynomial ring possesses the stable Betti numbers if and only if the ideal is componentwise linear was established. Such the theorem guarantees that componentwise linear ideals will play an important role in computational commutative algebra. Third, in order to obtain sophisticated generalization of Kruskal-Katona theorem in classical combinatorics on finite sets, via the discussion on the existence of a squarefree strongly stable ideal having the same graded Betti numbers as those of the generic initial ideal of a squarefree ideal in the polynomial ring, we did succeed in obtaining the affirmative answer to the outstanding conjecture that the graded Betti numbers of a squarefree ideal with a fixed Hubert function are less than or equal to those of the lexsegment ideal.
Social Activities
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なし
Devising educational methods
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1セメスターに少なくとも3回程度、レポート課題を課している。これは、「提出することに意味がある」ものではなく、丁寧に添削し、解答が間違っている場合はヒントや正解を書き込み、返却している。
Teaching materials
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教科書は、市販のものを使用している（所属学科のスタッフが作成した教科書も使っているが、筆者の赴任前なので、自分は執筆に関わっていいない）。自学科向けの専門科目では、教科書に載っていない題材を教えたい場合、自分で教科書・副読本風の教材を作成し、学生にプリント配布することもある。
Teaching method presentations
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特になし
Special notes on other educational activities
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特になし