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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Finiteness of cousin cohomologies


Author: Takesi Kawasaki
Journal: Trans. Amer. Math. Soc. 360 (2008), 2709-2739
MSC (2000): Primary 13D03; Secondary 13A30, 13F40, 13H10
Published electronically: September 25, 2007
MathSciNet review: 2373331
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04418-2
PII: S 0002-9947(07)04418-2
Keywords: Arithmetic Macaulayfication, Cohen-Macaulay ring, Cousin complex, excellent ring
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)
Article copyright: © Copyright 2007 American Mathematical Society