Torsion functors of local cohomology modules for complexes

ZHANG Pinger; MA Yajun

doi:10.13471/j.cnki.acta.snus.ZR20240278

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Torsion functors of local cohomology modules for complexes

Research articles | 更新时间：2025-11-14

- Torsion functors of local cohomology modules for complexes
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 64, Issue 4, Pages: 109-114(2025)
- 作者机构：
  
  1.甘肃政法大学人工智能学院，甘肃兰州 730070
  2.兰州交通大学数理学院，甘肃兰州 730070
- 作者简介：
  
  ZHANG Pinger（zpe7445@gsupl.edu.cn）
  MA Yajun（yjma@mail.lzjtu.cn）
- 基金信息：
- DOI：10.13471/j.cnki.acta.snus.ZR20240278
  CLC： O154.2
- Received：10 September 2024，
  
  Accepted：22 February 2025，
  
  Published Online：27 April 2025，
  
  Published：25 July 2025
- 稿件说明：
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张平儿,马亚军.复形的局部上同调模的torsion函子[J].中山大学学报(自然科学版)(中英文),2025,64(04):109-114.

ZHANG Pinger,MA Yajun.Torsion functors of local cohomology modules for complexes[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(04):109-114.
张平儿,马亚军.复形的局部上同调模的torsion函子[J].中山大学学报(自然科学版)(中英文),2025,64(04):109-114. DOI： 10.13471/j.cnki.acta.snus.ZR20240278.

ZHANG Pinger,MA Yajun.Torsion functors of local cohomology modules for complexes[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(04):109-114. DOI： 10.13471/j.cnki.acta.snus.ZR20240278.

摘要

证明了下有界复形的局部上同调模的torsion模属于Serre类，并且在某种情形下证明了复形的torsion模与局部上同调模的torsion模的同构式. 作为此同构式的应用，我们用复形的局部上同调模表示了复形的Betti数.

Abstract

We show that the torsion module

</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow></mfenced></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835944&type=

4.48733330

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835978&type=

26.50066566

is in a Serre subcategory for the bounded below

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835967&type=

2.62466669

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835968&type=

2.37066650

-complex

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835947&type=

2.62466669

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835956&type=

2.53999996

. In addition

we prove the isomorphism

</mo><mtext> </mtext><mi>X</mi></mrow></mfenced><mo>≅</mo><mi mathvariant="normal">T</mi><mi mathvariant="normal">o</mi><msubsup><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>R</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>R</mi><mo>/</mo><mi mathvariant="normal">a</mi><mo>

</mo><mtext> </mtext><msubsup><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow></mfenced></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835982&type=

4.48733330

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835971&type=

52.40866470

in some case. As an application

the Betti number of a complex

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835947&type=

2.62466669

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835956&type=

2.53999996

in a prime ideal

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835958&type=

3.38666677

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835984&type=

1.86266661

can be computed by the Betti number of the local cohomology modules of

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835947&type=

2.62466669

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835956&type=

2.53999996

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835974&type=

3.38666677

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=84835997&type=

1.86266661

关键词

Keywords

references

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DIBAEI M T ， VAHIDI A ， 2011 . Torsion functors of local cohomology modules ［J］. Algebr Represent Theor ， 14 ： 79 - 85 .

FOXBY H B ， IYENGAR S B ， 2001 . Depth and amplitude for unbounded complexes ［M］//AVRAMOV L L， et al. Commutative algebra： Interactions with algebraic geometry ， Providence ： American Mathematical Society .

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KHASHYARMANESH K ， 2007 . On the finiteness properties of extension and torsion functors of local cohomology modules ［J］. Proc Amer Math Soc ， 135 （ 5 ）： 1319 - 1327 .

MELKERSSON L ， 2005 . Modules cofinite with respect to an ideal ［J］. J Algebra ， 285 （ 2 ）： 649 - 668 .

RAHIMPOUR NASBI Y ， VAHIDI A ， AHMADI AMOLI K ， 2017 . Torsion functors of generalized local cohomology modules ［J］. Commun Algebra ， 45 （ 12 ）： 5420 - 5430 .

ROTMAN J ， 2009 . An introduction to homological algebra ［M］. New York ： Springer .

SCHENZEL P ， 2003 . Proregular sequences， local cohomology， and completion ［J］. Math Scand ， 92 （ 2 ）： 161 - 180 .

TARRÍO L A ， LÓPEZ A J ， LIPMAN J ， 1997 . Local homology and cohomology on schemes ［J］. Ann Sci de l'École Norm Supérieure ， 30 （ 1 ）： 1 - 39 .

YANG X Y ， LUO P ， 2025 . Local cohomology and Serre class in derived category D（ R ）［J］. Algebra Colloq ， 32 （ 1 ）： 41 - 52 .

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