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On the arithmetic of tight closure

Brenner, H. and Katzman, M. (2004) On the arithmetic of tight closure. Journal of the American Mathematical Society, 19 (3). pp. 659-672. ISSN 0894-0347


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We provide a negative answer to an old question in tight closure theory by showing that the containment x^3y^3 \in (x^4,y^4,z^4)^* in K[x,y,z]/(x^7+y^7-z^7) holds for infinitely many but not for almost all prime characteristics of the field K. This proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic. The ideal (x,y,z) \subset K[x,y,z,u,v,w]/(x^7+y^7-z^7, ux^4+vy^4+wz^4+x^3y^3) has then the property that the cohomological dimension fluctuates arithmetically between 0 and 1.

Item Type: Article
Copyright, Publisher and Additional Information: © 2006 American Mathematical Society. Reproduced in accordance with the publisher's self-archiving policy.
Keywords: tight closure; dependence on prime numbers; cohomological dimension; semistable bundles
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 18 Nov 2009 15:46
Last Modified: 16 Nov 2015 16:12
Published Version: http://www.ams.org/jams/2006-19-03/S0894-0347-05-0...
Status: Published
Publisher: American Mathematical Society
Refereed: Yes
URI: http://eprints.whiterose.ac.uk/id/eprint/10170

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