#### 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
##### 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 ${M}_{\mathfrak{m}}$ has finite $G$-dimension over ${R}_{\mathfrak{m}}$ for each maximal ideal $\mathfrak{m}$ 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