Vol. 4, No. 1, 2010

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Reflexivity and rigidity for complexes, I: Commutative rings

Luchezar L. Avramov, Srikanth B. Iyengar and Joseph Lipman

Vol. 4 (2010), No. 1, 47–86

A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main results characterizes C-rigid complexes. Specialized to the case when C is the relative dualizing complex of a homomorphism of rings of finite Gorenstein dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the way, new results about derived reflexivity with respect to C are established. Noteworthy is the statement that derived C-reflexivity is a local property; it implies that a finite R-module M has finite G-dimension over R if Mm has finite G-dimension over Rm for each maximal ideal m of R.

To our friend and colleague, Hans-Bjørn Foxby.

semidualizing complexes, perfect complexes, invertible complexes, rigid complexes, relative dualizing complexes, derived reflexivity, finite Gorenstein dimension
Mathematical Subject Classification 2000
Primary: 13D05, 13D25
Secondary: 13C15, 13D03
Received: 15 April 2009
Revised: 15 July 2009
Accepted: 18 August 2009
Published: 14 January 2010
Luchezar L. Avramov
Department of Mathematics
University of Nebraska
Lincoln, NE 68588
United States
Srikanth B. Iyengar
Department of Mathematics
University of Nebraska
Lincoln, NE 68588
United States
Joseph Lipman
Department of Mathematics
Purdue University
W. Lafayette, IN 47907
United States