On Multilinearity and Skew-symmetry of Certain Symbols in Motivic Cohomology of Fields

The purpose of the present article is to show the multilinearity for symbols in Goodwillie-Lichtenbaum motivic complex in two cases. The first case shown is where the degree is equal to the weight. In this case, the motivic cohomology groups of a field are isomorphic to the Milnor’s K-groups as shown by Nesterenko-Suslin, Totaro and Suslin-Voevodsky for various motivic complexes, but we give an explicit isomorphism for Goodwillie-Lichtenbaum complex in a form which visibly carries multilinearity of Milnor’s symbols to our multilinearity of motivic symbols. Next, we establish multilinearity and skew-symmetry for irreducible Goodwillie-Lichtenbaum symbols in Hl−1 M ` Spec k, Z(l) ́ . These properties have been expected to hold from the author’s construction of a bilinear form of dilogarithm in case k is a subfield of C and l = 2. The multilinearity of symbols may be viewed as a generalization of the well-known formula det(AB) = det(A) det(B) for commuting matrices.