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Transactions of the American Mathematical Society

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

 
 

 

Content of local cohomology, parameter ideals, and robust algebras


Authors: Melvin Hochster and Wenliang Zhang
Journal: Trans. Amer. Math. Soc. 370 (2018), 7789-7814
MSC (2010): Primary 13D45
DOI: https://doi.org/10.1090/tran/7226
Published electronically: July 31, 2018
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Abstract: This paper continues the investigation of quasilength, of content of local cohomology with respect to generators of the support ideal, and of robust algebras begun in joint work of Hochster and Huneke. We settle several questions raised by Hochster and Huneke. In particular, we give a family of examples of top local cohomology modules both in equal characteristic 0 and in positive prime characteristic that are nonzero but have content 0. We use the notion of a robust forcing algebra (the condition turns out to be strictly stronger than the notion of a solid forcing algebra in, for example, equal characteristic 0) to define a new closure operation on ideals. We prove that this new notion of closure coincides with tight closure for ideals in complete local domains of positive characteristic, which requires proving that forcing algebras for instances of tight closure are robust, and study several related problems. This gives, in effect, a new characterization of tight closure in complete local domains of positive characteristic. As a byproduct, we also answer a question of Lyubeznik in the negative.


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Additional Information

Melvin Hochster
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: hochster@umich.edu

Wenliang Zhang
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, Illinois 60607
Email: wlzhang@uic.edu

DOI: https://doi.org/10.1090/tran/7226
Received by editor(s): October 7, 2016
Received by editor(s) in revised form: February 8, 2017
Published electronically: July 31, 2018
Additional Notes: The first author was partially supported by the National Science Foundation through grant DMS#1401384.
The second author was partially supported by National Science Foundation through grant DMS#1606414
Article copyright: © Copyright 2018 American Mathematical Society

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