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On
the (non)rigidity of the Frobenius endomorphism over
Gorenstein rings
Hailong Dao, Jinjia Li and Claudia Miller |
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Vol. 4 (2010), No. 8, 1039–1053
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Abstract |
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It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this property to study the structure of such rings. One of our results states that the Picard group of the punctured spectrum of such a ring cannot have -torsion. When is a local complete intersection, this recovers (with a purely local algebra proof) an analogous statement for complete intersections in projective spaces first given by Deligne in SGA and also a special case of a conjecture by Gabber. Our method also leads to many simply constructed examples where rigidity for the Frobenius endomorphism does not hold, even when the rings are Gorenstein with isolated singularity. This is in stark contrast to the situation for complete intersection rings. A related length criterion for modules of finite length and finite projective dimension is discussed towards the end. |
Keywords
Frobenius endomorphism, rigidity, Tor, Picard group,
isolated singularity
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Mathematical Subject Classification 2000
Primary: 13A35
Secondary: 13D07, 14A05, 13C20
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Milestones
Received: 11 September 2009
Revised: 9 May 2010
Accepted: 11 June 2010
Published: 24 February 2011
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Authors |
| Hailong Dao | |
| Department of Mathematics University of Kansas Lawrence, KS 66045-7523 United States |
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| Jinjia Li | |
| Department of Mathematical
Sciences Middle Tennessee State University Murfreesboro, TN 37132 United States |
Department of Mathematics University of Louisville 328 Natural Sciences Building Louisville, KY 40292 United States |
| Claudia Miller | |
| Mathematics Department Syracuse University Syracuse, NY 13244-1150 United States |
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