TY - JOUR

T1 - Descent via Koszul extensions

AU - Christensen, Lars Winther

AU - Sather-Wagstaff, Sean

N1 - Funding Information:
* Corresponding author at: Department of Mathematics, 300 Minard Hall, North Dakota State University, Fargo, ND 58105-5075, USA. E-mail addresses: lars.w.christensen@math.ttu.edu (L.W. Christensen), sean.sather-wagstaff@ndsu.edu (S. Sather-Wagstaff). URLs: http://www.math.ttu.edu/~lchriste (L.W. Christensen), http://math.ndsu.nodak.edu/faculty/ssatherw/ (S. Sather-Wagstaff). 1 This work was done while L.W.C. visited University of Nebraska–Lincoln, partly supported by grants from the Danish Natural Science Research Council and the Carlsberg Foundation. 2 S.S.-W. was partially supported by NSF grant NSF 0354281.

PY - 2009/11/1

Y1 - 2009/11/1

N2 - Let R be a commutative noetherian local ring with completion over(R, ̂). We apply differential graded (DG) algebra techniques to study descent of modules and complexes from over(R, ̂) to R′ where R′ is either the henselization of R or a pointed étale neighborhood of R: We extend a given over(R, ̂)-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 over(R, ̂) 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 over(R, ̂) when R is excellent, henselian, and Cohen-Macaulay.

AB - Let R be a commutative noetherian local ring with completion over(R, ̂). We apply differential graded (DG) algebra techniques to study descent of modules and complexes from over(R, ̂) to R′ where R′ is either the henselization of R or a pointed étale neighborhood of R: We extend a given over(R, ̂)-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 over(R, ̂) 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 over(R, ̂) when R is excellent, henselian, and Cohen-Macaulay.

KW - Artin approximation

KW - Descent

KW - Koszul extensions

KW - Liftings

KW - Semi-dualizing complexes

KW - Semidualizing

UR - http://www.scopus.com/inward/record.url?scp=70349394403&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2008.03.007

DO - 10.1016/j.jalgebra.2008.03.007

M3 - Article

AN - SCOPUS:70349394403

VL - 322

SP - 3026

EP - 3046

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 9

ER -