Motivic cohomology (definition)
Statement
For smooth schemes over a field (and in Voevodsky’s triangulated categories of effective motives), motivic cohomology is a family of bigraded abelian groups H^{p,q}(X, ℤ(n)) (weight q, twist n), functorial in X, built from motivic complexes of sheaves with transfers on the Nisnevich site. The theory is designed to satisfy the Bloch–Ogus axioms for a “universal” twisted cohomology theory relating to algebraic cycles and K-theory; comparison with Bloch higher Chow groups is a separate identification theorem (motivic-cohomology-higher-chow). Constructions, functoriality, and operations are proved in algebraic geometry; formalizable in ZFC once schemes and sheaves are encoded.
Formula
H^{p,q}(X, ℤ(n)) (smooth X / motives); Bloch–Ogus package
Equation
Motivic complexes; sheaves with transfers; Nisnevich hypercohomology
Proof sketch
Voevodsky: triangulated category DM_eff(k) of effective motives; motivic complexes ℤ(n) and their hypercohomology define H^{*,*}(−, ℤ(n)). Foundational “Homology of schemes” (Selecta Math., 1996) and the motivic stable homotopy setup (see also theorem a1-homotopy-theory for 𝔸¹-local homotopy).
Source
Voevodsky, V. Homology of schemes, Selecta Mathematica (N.S.) 2 (1996). See also Friedlander, Suslin, Voevodsky, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud. 143, Princeton Univ. Press, 2000.
Domains
mathematics