On multilinearity and skew-symmetry of certain symbols

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 $H^{l-1}_{\M} \bigl(\Spec k , \Z(l) \bigr)$. 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.

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Published 1 January 2009