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Arithmetical rank and projective dimension of powers of Stanley-Reisner ideals

Research Project

Project/Area Number 18K03244
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionOkayama University (2019-2023)
Saga University (2018)

Principal Investigator

TERAI Naoki  岡山大学, 環境生命自然科学学域, 教授 (90259862)

Co-Investigator(Kenkyū-buntansha) 木村 杏子  静岡大学, 理学部, 准教授 (60572633)
吉田 健一  日本大学, 文理学部, 教授 (80240802)
宮崎 誓  熊本大学, 大学院先端科学研究部(理), 教授 (90229831)
Project Period (FY) 2018-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
KeywordsStanley-Reisner ring / simplicial complex / edge ideal / arithmetical rank / projective dimension / depth / Serre' s condition / local cohomology / edge ideal / simplicial complex / very well-covered / regularity / edge-weighted / Cohen-Macaulay / unmixed / Stanley-Reisner ideal / second symbolic power / binomial edge ideal / licci / second power / Stanley-Reisner イデアル / Gorenstein
Outline of Final Research Achievements

We gave a classification of Stanley-Reisner rings of codimension two with some higher Serre index. By using this classification, we show the equality of projective dimension of the Stanley-Reisner rings and the arithmetical rank of their Stanley-Reisner ideals. Moreover, our classification allows us to compute the h-vectors and give a negative answer to some question regarding to these vectors.
We showed that third or more ordinary or symbolic powers of Stanley-Reisner ideals are level if and only if they are equi-generated complete intersections.

Academic Significance and Societal Importance of the Research Achievements

私が研究している可換環論は代数学の基礎をなす学問のひとつで代数幾何、整数論の基礎的なツールであるとともに可換環独自の問題意識からも今日も活発に研究されている。私は組合せ論的可換環論という組合せ論を用いて多項式内の単項式イデアルを調べるという可換環論を研究している。この分野は計算機の進歩に伴ってした側面があり、それらとの親和性も強い。本研究は可換環論的方法のみでは扱えなかった問題に対して組合せ論的、位相幾何学的なアプローチも用いて問題解決を目指した基礎研究である。

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (39 results)

All 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (12 results) Journal Article (14 results) (of which Int'l Joint Research: 14 results,  Peer Reviewed: 14 results,  Open Access: 2 results) Presentation (13 results) (of which Int'l Joint Research: 5 results,  Invited: 9 results)

  • [Int'l Joint Research] Sharif 工業大学(イラン)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Messina 大学(イタリア)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Purdue 大学(米国)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of Tehran/Sharif University of Technology(イラン)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] University of Messina(イタリア)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Ovidius University(ルーマニア)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] University of Tehran(イラン)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] University of Trento(イタリア)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Ovidius University(ルーマニア)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] University of Trento(イタリア)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Institute of Mathematics/Hanoi National University of Education(ベトナム)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] トレント大学(イタリア)

    • Related Report
      2018 Research-status Report
  • [Journal Article] On the dimension of dual modules of local cohomology and the Serre's condition for the unmixed Stanley-Reisner ideals of small height2023

    • Author(s)
      Pournaki M.R.、Poursoltani M.、Terai N.、Yassemi S.
    • Journal Title

      Journal of Algebra

      Volume: 632 Pages: 751-782

    • DOI

      10.1016/j.jalgebra.2023.05.031

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Licci level Stanley-Reisner ideals with height three2023

    • Author(s)
      Rinaldo Giancarlo、Terai Naoki
    • Journal Title

      Sao Paulo Journal of Mathematical Sciences

      Volume: 17 Issue: 1 Pages: 345-386

    • DOI

      10.1007/s40863-022-00326-8

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Simplicial Complexes Satisfying Serre's Condition versus the Ones Which Are Cohen--Macaulay in a Fixed Codimension2022

    • Author(s)
      Pournaki M. R.、Poursoltani M.、Terai N.、Yassemi S.
    • Journal Title

      SIAM Journal on Discrete Mathematics

      Volume: 36 Issue: 4 Pages: 2506-2522

    • DOI

      10.1137/21m1439687

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A note on monomial ideals which are Cohen?Macaulay in a fixed codimension2022

    • Author(s)
      Pournaki M. R.、Shibata K.、Terai N.、Yassemi S.
    • Journal Title

      Communications in Algebra

      Volume: 50 Issue: 11 Pages: 4988-4996

    • DOI

      10.1080/00927872.2022.2079663

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Sequentially Cohen?Macaulay binomial edge ideals of closed graphs2022

    • Author(s)
      Ene Viviana、Rinaldo Giancarlo、Terai Naoki
    • Journal Title

      Research in the Mathematical Sciences

      Volume: 9 Issue: 3

    • DOI

      10.1007/s40687-022-00334-2

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Very well-covered graphs and local cohomology of their residue rings by the edge ideals2022

    • Author(s)
      Kimura K.、Pournaki M.R.、Terai N.、Yassemi S.
    • Journal Title

      Journal of Algebra

      Volume: 606 Pages: 1-18

    • DOI

      10.1016/j.jalgebra.2022.04.021

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra2022

    • Author(s)
      Kimura K.、Pournaki M. R.、Seyed Fakhari S. A.、Terai N.、Yassemi S.
    • Journal Title

      Research in the Mathematical Sciences

      Volume: 9 Issue: 2

    • DOI

      10.1007/s40687-022-00326-2

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs2021

    • Author(s)
      Seyed Fakhari Seyed Amin、Shibata Kosuke、Terai Naoki、Yassemi Siamak
    • Journal Title

      Communications in Algebra

      Volume: 49 Issue: 10 Pages: 4249-4257

    • DOI

      10.1080/00927872.2021.1917590

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Licci Level Stanley-Reisner Ideals with Height Three and with Type Two2020

    • Author(s)
      Rinaldo Giancarlo、Terai Naoki、Yoshida Ken-Ichi
    • Journal Title

      Combinatorial Structures in Algebra and Geometry(Springer Proceedings in Mathematics & Statistics

      Volume: 331 Pages: 123-142

    • DOI

      10.1007/978-3-030-52111-0_10

    • ISBN
      9783030521103, 9783030521110
    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Licci binomial edge ideals2020

    • Author(s)
      Ene Viviana、Rinaldo Giancarlo、Terai Naoki
    • Journal Title

      Journal of Combinatorial Theory, Series A

      Volume: 175 Pages: 105278-105278

    • DOI

      10.1016/j.jcta.2020.105278

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] 4-Dimensional Licci Gorenstein Stanley-Reisner Ideals2019

    • Author(s)
      Rinaldo Giancarlo、Terai Naoki
    • Journal Title

      Acta Mathematica Vietnamica

      Volume: 44 Issue: 3 Pages: 691-700

    • DOI

      10.1007/s40306-019-00339-0

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Cohen-Macaulay and (S2) Properties of the Second Power of Squarefree Monomial Ideals2019

    • Author(s)
      Hoang Do Trong、Rinaldo Giancarlo、Terai Naoki
    • Journal Title

      Mathematics

      Volume: 7 Issue: 8 Pages: 684-684

    • DOI

      10.3390/math7080684

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Level property of ordinary and symbolic powers of Stanley-Reisner ideals2019

    • Author(s)
      Minh Nguyen Cong、Terai Naoki、Thuy Phan Thi
    • Journal Title

      Journal of Algebra

      Volume: 535 Pages: 350-364

    • DOI

      10.1016/j.jalgebra.2019.05.044

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Depth and regularity modulo a principal ideal2018

    • Author(s)
      Caviglia Giulio、Ha Huy Tai、Herzog Jurgen、Kummini Manoj、Terai Naoki、Trung Ngo Viet
    • Journal Title

      Journal of Algebraic Combinatorics

      Volume: 49 Issue: 1 Pages: 1-20

    • DOI

      10.1007/s10801-018-0811-9

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Level Stanley-Reisner rings with codimension two2023

    • Author(s)
      Naoki Terai
    • Organizer
      Commutative Algebra and its interaction with Algebraic Geometry and Combinatorics 2023
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 2-Cohen-Macauly Stanley-Reisner rings with codimension two2023

    • Author(s)
      Naoki Terai
    • Organizer
      第44 回可換環論シンポジウム
    • Related Report
      2023 Annual Research Report
  • [Presentation] 余次元の小さいStanley-Reisner環について2022

    • Author(s)
      寺井直樹
    • Organizer
      東京可換セミナー
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Stanley--Reisner rings with low codimension2022

    • Author(s)
      寺井直樹
    • Organizer
      第43回可換環論シンポジウム
    • Related Report
      2022 Research-status Report
  • [Presentation] The dual modules of the local cohomology of Stanley-Reisner rings with low codimension2022

    • Author(s)
      寺井直樹
    • Organizer
      Algebra & Number Theory Seminar  Institute of Mathematics, Hanoi, Vietnam
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Cohen-Macaulay property of weighted edge ideals of very well-covered graphs2021

    • Author(s)
      寺井直樹
    • Organizer
      IIT Bombay Virtual Commutative Algebra Seminar
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Edge-weighted edge ideals of very well-covered graphs2020

    • Author(s)
      柴田孝祐, 寺井直樹
    • Organizer
      オンライン研究集会 組合せ論と可換環論
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Cohen-Macaulay property of edge-weighted very well-covered graphs2020

    • Author(s)
      寺井直樹
    • Organizer
      One day workshop on commutative algebra and related fields
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Cohen-Macaulay and (S2) properties of the second power of squarefree monomial ideals2019

    • Author(s)
      寺井直樹
    • Organizer
      Vietnam - US Joint Mathematical Meeting
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Licci level Stanley-Reisner rings with codimension 3 and with Cohen-Macaulay type 2,2019

    • Author(s)
      寺井直樹
    • Organizer
      第32回可換環論セミナー
    • Related Report
      2019 Research-status Report
  • [Presentation] Licci level Stanley-Reisner rings with codimension three and with Cohen-Macaulay type two2019

    • Author(s)
      寺井直樹
    • Organizer
      One day workshop on commutative algebra and related fields
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Licci level Stanely-Reisner ideal について2019

    • Author(s)
      寺井直樹
    • Organizer
      組合せ論と可換代数オータムセミナー
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 4-dimensional Gorenstein licci Stanley-Reisner ideals2018

    • Author(s)
      寺井直樹
    • Organizer
      可換環論セミナー
    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2025-01-30  

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