Vol. 4, No. 8, 2010

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings

Hailong Dao, Jinjia Li and Claudia Miller

Vol. 4 (2010), No. 8, 1039–1053
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.

Keywords
Frobenius endomorphism, rigidity, Tor, Picard group, isolated singularity
Mathematical Subject Classification 2000
Primary: 13A35
Secondary: 13D07, 14A05, 13C20
Milestones
Received: 11 September 2009
Revised: 9 May 2010
Accepted: 11 June 2010
Published: 24 February 2011
Authors
Hailong Dao
Department of Mathematics
University of Kansas
Lawrence, KS 66045-7523
United States
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