Reflexivity and rigidity for complexes, I: Commutative rings

Avramov, Luchezar; Iyengar, Srikanth; Lipman, Joseph

doi:10.2140/ant.2010.4.47

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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

DOI: 10.2140/ant.2010.4.47

Abstract

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 $M_{m}$ has finite $G$ -dimension over $R_{m}$ for each maximal ideal $m$ of $R$ .

To our friend and colleague, Hans-Bjørn Foxby.

Keywords

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

Milestones

Received: 15 April 2009

Revised: 15 July 2009

Accepted: 18 August 2009

Published: 14 January 2010

Authors

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
http://www.math.unl.edu/~siyengar2
Joseph Lipman
Department of Mathematics Purdue University W. Lafayette, IN 47907 United States

Vol. 4, No. 1, 2010