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2010 On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings
Hailong Dao, Jinjia Li, Claudia Miller
Algebra Number Theory 4(8): 1039-1053 (2010). DOI: 10.2140/ant.2010.4.1039

Abstract

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 R cannot have p-torsion. When R 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.

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Hailong Dao. Jinjia Li. Claudia Miller. "On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings." Algebra Number Theory 4 (8) 1039 - 1053, 2010. https://doi.org/10.2140/ant.2010.4.1039

Information

Received: 11 September 2009; Revised: 9 May 2010; Accepted: 11 June 2010; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1221.13004
MathSciNet: MR2832633
Digital Object Identifier: 10.2140/ant.2010.4.1039

Subjects:
Primary: 13A35
Secondary: 13C20 , 13D07 , 14A05

Keywords: Frobenius endomorphism , isolated singularity , Picard group , rigidity , Tor

Rights: Copyright © 2010 Mathematical Sciences Publishers

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