Mathematics – K-Theory and Homology
Scientific paper
2008-01-09
Math. Res. Lett. 16 (2009): 303-322
Mathematics
K-Theory and Homology
We acknowledge that mutilinearity does not hold for arbitrary symbols in H^{l-1} (Spec k, Z(l)) as claimed in the 1st version
Scientific paper
The purpose of the present article is to show the multilinearity for symbols in Goodwillie-Lichtenbaum 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} (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 the field of complex numbers and l=2. Next, we establish multilinearity and skew-symmetry for Goodwillie-Lichtenbaum symbols in H^{l-1} (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 the field of complex numbers 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 tuples of commuting matrices.
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