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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Finiteness of cousin cohomologies
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by Takesi Kawasaki PDF
Trans. Amer. Math. Soc. 360 (2008), 2709-2739 Request permission

Abstract:

The notion of the Cousin complex of a module was given by Sharp in 1969. It wasn’t known whether its cohomologies are finitely generated until recently. In 2001, Dibaei and Tousi showed that the Cousin cohomologies of a finitely generated $A$-module $M$ are finitely generated if the base ring $A$ is local, has a dualizing complex, $M$ satisfies Serre’s $(S_2)$-condition and is equidimensional. In the present article, the author improves their result. He shows that the Cousin cohomologies of $M$ are finitely generated if $A$ is universally catenary, all the formal fibers of all the localizations of $A$ are Cohen-Macaulay, the Cohen-Macaulay locus of each finitely generated $A$-algebra is open and all the localizations of $M$ are equidimensional. As a consequence of this, he gives a necessary and sufficient condition for a Noetherian ring to have an arithmetic Macaulayfication.
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Additional Information
  • Takesi Kawasaki
  • Affiliation: Department of Mathematics, Tokyo Metropolitan University, Hachioji-shi Minami Ohsawa 1-1,d Tokyo, 192-0397, Japan
  • Email: kawasaki@tmu.ac.jp
  • Received by editor(s): October 8, 2003
  • Received by editor(s) in revised form: September 4, 2006
  • Published electronically: September 25, 2007
  • Additional Notes: This work was supported by the Japan Society for the Promotion of Science (the Grant-in-Aid for Scientific Research (C)(2) 13640034)
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 2709-2739
  • MSC (2000): Primary 13D03; Secondary 13A30, 13F40, 13H10
  • DOI: https://doi.org/10.1090/S0002-9947-07-04418-2
  • MathSciNet review: 2373331