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

摘要

We construct for any scheme X with a dualizing complex I center dot a Gersten-Witt complex GW(X, I center dot) 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 GrGWp(X, I center dot) of this filtration we prove H-P(GrGW (p)(X, I center dot)) similar or equal to CHp(X, mu(I))/2, where mu(I) is the codimension function of the dualizing complex I center dot and CHp(X, mu(I)) denotes the Chow group of mu(I)-codimension p-cycles modulo rational equivalence. (c) 2006 Elsevier B.V. All rights reserved.