Descent of Semidualizing Complexes for Rings with the Approximation Property

Let R be a commutative noetherian local ring with completion b R. When R has the approximation property, we prove an approximation result for complexes with finitely generated homology. This is used to investigate descent of semidualizing complexes from b R to R. We show that, if R has the approximation property, then there is a bijective correspondence between semidualizing b R-complexes and semidualizing R-complexes. In particular, we recover a result of Hinich and Rotthaus stating that every ring with the approximation property has a dualizing complex. As an application of the descent theorem, we prove a new version of a classical result on uniform annihilation of homology modules of perfect complexes. Finally, we resolve the finiteness question for the set of isomorphism classes of semidualizing R-modules, when R is Cohen–Macaulay and equicharacteristic.