On the Grayson Spectral Sequence

One can use next the globalization machinery developed in [S-F] to get a similar looking spectral sequence for any smooth scheme of finite type over a field. Moreover, it’s not hard to see that the resulting spectral sequence coincides with the one constructed in [F-S]. What is nice, however, with this approach is that it avoids completely the use of the paper of Bloch and Lichtenbaum [B-L], which many people still find doubtful, still is not published and possibly never will be. The main result of the paper says that the canonical homomorphism of complexes of sheaves Z(n) −→ Z(n) is a quasi-isomorphism. There are essentially three reasons behind this quasi-isomorphism. First, cohomology sheaves of the complex Z(n) are homotopy invariant K 0 -sheaves and hence homotopy invariant pretheories M. Walker [W]. Second, the complex Z(n) is defined by a rationally contractible presheaf, as is the complex Z(n); see Proposition 2.2 below, which implies vanishing of certain polyrelative cohomology groups in the semilocal case as in Theorem 2.7 below. Finally, just as the usual motivic cohomology does, Grayson’s motivic cohomology satisfies cohomology purity, i.e., if Z ⊂ Y is a smooth subscheme of pure codimension d, then we have canonical Gysin isomorphisms