Reflexivity and rigidity for complexes. I. Commutative rings

TL;DR

This paper introduces and studies a new notion of rigidity for complexes over commutative noetherian rings, generalizing existing concepts and establishing key properties of C-rigid complexes and derived reflexivity.

Contribution

It defines C-rigidity for complexes, characterizes C-rigid complexes, and extends results on rigid dualizing complexes to a broader context involving semidualizing complexes.

Findings

C-rigid complexes are characterized explicitly.

Derived C-reflexivity is a local property.

Finite G-dimension is implied by local G-dimension finiteness.

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 result 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, a number of new results concerning 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 it is locally of finite G-dimension.