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Research on rational singularities and almost Gorenstein blow-up algebras

Research Project

Project/Area Number 16K05110
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionNihon University

Principal Investigator

YOSHIDA Ken-ichi  日本大学, 文理学部, 教授 (80240802)

Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2018: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords概Gorenstein環 / pg イデアル / good イデアル / 有理特異点 / Ulrich イデアル / strong Rees property / normal reduction number / almost Gorenstein ring / pg ideal / 概ゴレンシュタイン / 整閉イデアル / Buchsbaum 加群 / Ulrich 加群 / Rees 代数 / 代数学 / 可換環論 / 特異点論
Outline of Final Research Achievements

We proved that if the Rees algebra of any parameter ideal in a Gorenstein local ring is an almost Gorenstein local ring then the ring is regular, and that the Rees algebra of an ideal generated by part of a regular system of parameters is an almost Gorenstein graded ring. We characterized the notion of pg ideals in terms of Rees algebras, and proved that the Rees algebra of any pg ideal is an almost Gorenstein local ring under suitable conditions.

Academic Significance and Societal Importance of the Research Achievements

我々は Rees 代数を通して, 「概 Gorenstein 環」と「概 Gorenstein 次数付き環」の概念の違いを浮き彫りにした。この研究はその後の概念を導入する際に参考になったと思われる。また,研究代表者らが本研究の前に導入した pg イデアルの概念を環論的に同値な条件に読み替えることにより, 有理特異点のイデアル論を超える形で, 2次元特異点に付随するイデアル論の道を開くことができたという事実は重要な成果であると言えるだろう。

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (25 results)

All 2019 2018 2017 2016 Other

All Journal Article (10 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 10 results,  Open Access: 10 results,  Acknowledgement Compliant: 3 results) Presentation (10 results) (of which Int'l Joint Research: 2 results,  Invited: 3 results) Remarks (2 results) Funded Workshop (3 results)

  • [Journal Article] A characterization of two-dimensional rational singularities via core of ideals2018

    • Author(s)
      Okuma Tomohiro、Watanabe Kei-ichi、Yoshida Ken-ichi
    • Journal Title

      J. Algebra

      Volume: 499 Pages: 450-468

    • DOI

      10.1016/j.jalgebra.2017.11.053

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Almost Gorensteinness of Rees algebras of p_g ideals2018

    • Author(s)
      Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi and Ken-ichi Yoshida
    • Journal Title

      Michigan Math.J.

      Volume: 67 Pages: 159-174

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] ON THE MOST EXPECTED NUMBER OF COMPONENTS FOR RANDOM LINKS2017

    • Author(s)
      Kazuhiro Ichihara and Ken-ichi Yoshida
    • Journal Title

      Tohoku Mathematical Journal, Second Series

      Volume: 69 Issue: 4 Pages: 637-641

    • DOI

      10.2748/tmj/1512183634

    • ISSN
      0040-8735, 2186-585X
    • Year and Date
      2017-12-30
    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Ulrich modules over cyclic quotient surface singularities2017

    • Author(s)
      Yusuke Nakajima and Ken-ichi Yoshida
    • Journal Title

      J. Algebra

      Volume: 482 Pages: 224-247

    • DOI

      10.1016/j.jalgebra.2017.03.018

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A Duality in Buchsbaum rings and triangulated manifolds2017

    • Author(s)
      Satoshi Murai, Isabella Novik and Ken-ichi Yoshida
    • Journal Title

      Algebra and Number Theory

      Volume: 11 Issue: 3 Pages: 635-656

    • DOI

      10.2140/ant.2017.11.635

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Rees algebras and $p_g$-ideals in a two-dimensional normal local domain2017

    • Author(s)
      T. Okuma,K. Watanabe and K. Yoshida
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: 145 Issue: 1 Pages: 39-47

    • DOI

      10.1090/proc/13235

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The almost Gorenstein Rees algebras over two-dimensional regular local rings2016

    • Author(s)
      Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, and Ken-ichi Yoshida
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 220 Issue: 10 Pages: 3425-3436

    • DOI

      10.1016/j.jpaa.2016.04.007

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] The almost Gorenstein Rees algebras of parameters2016

    • Author(s)
      Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, and Ken-ichi Yoshida
    • Journal Title

      Journal of Algebra

      Volume: 452 Pages: 263-278

    • DOI

      10.1016/j.jalgebra.2015.12.022

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Good ideals and $p_g$-ideals in a two-dimensional normal singularities2016

    • Author(s)
      T.Okuma, K. Watanabe, K. Yoshida
    • Journal Title

      Manuscripta mathematica

      Volume: 未定 Issue: 3-4 Pages: 499-520

    • DOI

      10.1007/s00229-016-0821-7

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Ulrich ideals and modules of two-dimensional rational singularities2016

    • Author(s)
      Shiro Goto; Kazuho Ozeki; Ryo Takahashi; Kei-ichi Watanabe; Ken-ichi Yoshida
    • Journal Title

      Nagoya Math. J.

      Volume: 221 Issue: 1 Pages: 69-110

    • DOI

      10.1017/nmj.2015.1

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Presentation] Lower bound of Hilbert-Kunz multiplicities2019

    • Author(s)
      吉田健一
    • Organizer
      AMS Sectional Meeting AMS Special Session, Special Session on Commutative Algebra and its Environs, I, University of Hawaii at Manoa, Honolulu
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Lower bound of Hilbert-Kunz multiplicity2018

    • Author(s)
      渡辺敬一・吉田健一
    • Organizer
      第40回可換環論シンポジウム
    • Related Report
      2018 Annual Research Report
  • [Presentation] Normal reduction numbers of some 2-dimensional normal hypersurfaces2018

    • Author(s)
      吉田健一
    • Organizer
      第19回岡山可換代数表現セミナー (OSCAR19)
    • Related Report
      2017 Research-status Report
  • [Presentation] Normal reduction numbers of integrally closed ideals2017

    • Author(s)
      吉田健一, 奥間智弘, 渡辺敬一
    • Organizer
      RIMS共同研究(公開型)第39可換環論シンポジウム
    • Related Report
      2017 Research-status Report
  • [Presentation] Strong Rees property for powers of the maximal ideal2017

    • Author(s)
      吉田健一, 奥間智弘, 渡辺敬一
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] Characteristic p>0 methods in commutative algebra2017

    • Author(s)
      吉田健一
    • Organizer
      International school and Workshop on Commutative Algebra by and for young mathematicians
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Dao's question and strong Rees property2017

    • Author(s)
      Ken-ichi Yoshida
    • Organizer
      第29回可換環論セミナー
    • Place of Presentation
      山口大学大学会館 (山口県)
    • Related Report
      2016 Research-status Report
  • [Presentation] Almost Gorenstein property of Rees algebras of powers of maximal ideals2016

    • Author(s)
      Ken-ichi Yoshida
    • Organizer
      Commutative Algebra Day in Tokyo
    • Place of Presentation
      東京大学 (東京都)
    • Related Report
      2016 Research-status Report
  • [Presentation] Introduction to Ulrich modules and ideals2016

    • Author(s)
      Ken-ichi Yoshida
    • Organizer
      RIMS研究集会, Non-commutative crepant resolutions, Ulrich modules and generalizations of the McKay correspondence
    • Place of Presentation
      京都大学数理解析研究所 (京都府)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometric genus of cone singularities2016

    • Author(s)
      Okuma Tomohiro, Kei-ichi Watanabe, Ken-ichi Yoshida
    • Organizer
      第38回可換環論シンポジウム
    • Place of Presentation
      IPC (神奈川県葉山市)
    • Related Report
      2016 Research-status Report
  • [Remarks] 日本大学文理学部

    • URL

      http://kenkyu-web.cin.nihon-u.ac.jp/scripts/websearch/index.htm

    • Related Report
      2017 Research-status Report
  • [Remarks] https://www.chs.nihon-u.ac.jp/research/

    • Related Report
      2016 Research-status Report
  • [Funded Workshop] The 40th Symposium on Commutative Algebra in Japan2018

    • Related Report
      2018 Annual Research Report
  • [Funded Workshop] Commutative Algebra and Related Topics2017

    • Related Report
      2017 Research-status Report
  • [Funded Workshop] RIMS研究集会, Non-commutative crepant resolutions, Ulrich modules and generalizations of the McKay correspondence2016

    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-06-13
    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2020-03-30  

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