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

We construct for any scheme X with a dualizing complex I• a Gersten–Witt complex image 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 image of this filtration we prove image, where μI is the codimension function of the dualizing complex I• and image denotes the Chow group of μI-codimension p-cycles modulo rational equivalence.