A graded Gersten–Witt complex for schemes with a dualizing complex and the Chow group

Abstract

We construct for any scheme X with a dualizing complex I • a Gersten–Witt complex GW ( X , I • ) and show that the differential of this complex respects the filtration by the powers of the fundamental ideal. To prove this we introduce second residue maps for one-dimensional local domains which have a dualizing complex. This residue maps generalize the classical second residue morphisms for discrete valuation rings. For the cohomology of the quotient complexes GrGW p ( X , I • ) of this filtration we prove H p ( GrGW p ( X , I • ) ) ≃ CH p ( X , μ I ) / 2 , where μ I is the codimension function of the dualizing complex I • and CH p ( X , μ I ) denotes the Chow group of μ I -codimension p -cycles modulo rational equivalence.