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Local Cohomology Annihilators and Macaulayfication

Abstract

The aim of this paper is to study a deep connection between local cohomology annihilators and Macaulayfication and arithmetic Macaulayfication over a local ring. Local cohomology annihilators appear through the notion of p-standard system of parameters. For a local ring, we prove an equivalence of the existence of Macaulayfications, the existence of a p-standard system of parameters, being a quotient of a Cohen-Macaulay local ring, and the verification of Faltings’ Annihilator theorem. For a finitely generated module which is unmixed and faithful, we prove an equivalence of the existence of an arithmetic Macaulayfication and the existence of a p-standard system of parameters; and both are proved to be equivalent to the existence of an arithmetic Macaulayfication on the ground ring. A connection between Macaulayfication and universal catenaricity is also discussed.

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Acknowledgments

The authors thank M. Brodmann and S. Goto for useful discussions and comments during this work. They thank the anonymous referee for some helpful comments on the earlier version. The second author thanks the Vietnam Institute for Advanced Study in Mathematics (VIASM), Vietnam, and the Institute for Mathematical Sciences (IMS-NUS), Singapore, for the support and hospitality during his visit to these institutions.

Nguyen Tu Cuong is supported by the NAFOSTED of Vietnam under grant number 101.04-2014.25.

Doan Trung Cuong is partially supported by the SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties” and by the NAFOSTED of Vietnam.

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Cuong, N.T., Cuong, D.T. Local Cohomology Annihilators and Macaulayfication. Acta Math Vietnam 42, 37–60 (2017). https://doi.org/10.1007/s40306-016-0185-9

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Keywords

  • Arithmetic Macaulayfication
  • Macaulayfication
  • Local cohomology annihilator
  • P-standard system of parameters
  • Quotient of Cohen-Macaulay ring

Mathematics Subject Classification (2010)

  • Primary 13H10
  • 14M05
  • Secondary 13D45
  • 14B05