Descent via Koszul extensions

Abstract

Let R be a commutative noetherian local ring with completion R ˆ . We apply differential graded (DG) algebra techniques to study descent of modules and complexes from R ˆ to R ′ where R ′ is either the henselization of R or a pointed étale neighborhood of R: We extend a given R ˆ -complex to a DG module over a Koszul complex; we describe this DG module equationally and apply Artin approximation to descend it to R ′ . This descent result for Koszul extensions has several applications. When R is excellent, we use it to descend the dualizing complex from R ˆ to a pointed étale neighborhood of R; this yields a new version of P. Roberts' theorem on uniform annihilation of homology modules of perfect complexes. As another application we prove that the Auslander Condition on uniform vanishing of cohomology ascends to R ˆ when R is excellent, henselian, and Cohen–Macaulay.