Proof — Motivic cohomology and Bloch higher Chow
Statement
For smooth schemes of finite type over a field, motivic cohomology—defined via motivic complexes of Nisnevich sheaves with transfers—is canonically isomorphic to Bloch’s higher Chow groups in the bidegrees established in the cycle-complex framework. (Standard indexing: for twist n and suitable p, one has H^{p,n}(X, ℤ(n)) ≅ CH^n(X, 2n − p), relating motivic weight to algebraic cycles.) This identification packages moving lemmas and higher algebraic cycles into motivic cohomology; it is distinct from the later ℤ/2 Milnor-conjecture paper. Proofs are in the FSVM cycle-book program; formalizable in ZFC.
Formula
H^{p,n}(X, ℤ(n)) ≅ CH^n(X, 2n − p) (smooth X over a field; standard grading)
Equation
Cycle complexes z^q(X, •); quasi-isomorphism with motivic complexes
Proof sketch
Friedlander, E. M.; Suslin, A.; Voevodsky, V. Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies 143, Princeton University Press, 2000. Part IV and related chapters establish the comparison between motivic cohomology and Bloch higher Chow / Suslin homology.
Source
Friedlander, Suslin, Voevodsky, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud. 143, Princeton Univ. Press, 2000.
Domains
mathematics