Motivic Cohomology of Pairs of Simplices
نویسنده
چکیده
Let F be a eld and n> 2 be a positive integer. A simplex in the projective space P F is an ordered set of hyperplanes L 1⁄4 ðL0; . . . ; LnÞ. A face of L is any nonempty intersection of the hyperplanes. A pair of simplices is admissible if they do not have common faces of the same dimension. It is a generic pair if all the faces of the two simplices are in general position. In their seminal paper [2, 4] Beilinson et al. initiated the study of the motivic cohomology of the admissible pairs of simplices ðL;M Þ on PF which is de ned as a graded comodule L2 j1⁄40H motðL;M Þj over a suitable graded algebra AðF Þ ;Q :1⁄4 AðF Þ Q. The cohomology is the arithmetico-algebraic analog of the relative Betti cohomology and is de ned via mysterious complexes formed by the geometric combinatorial data of the pair. The reason for this is that the graded pieces of the cohomology groups can actually be obtained from a spectral sequence converging to H ðPC n L;M n L;QÞ with degenerate E2-term. In this paper we propose a generalization of the above from PF to P n F . Brie:y speaking, the group AnðF Þ is generated by admissible pairs of simplices in P F , subject to a set of relations (see De nition 2.1 for the detail). The de ning (double scissors congruence) relations re:ect (conjecturally all of) the functional equations of Aomoto polylogarithms rst studied in [1]. These groups are expected to form a Hopf algebra and are closely related to algebraic K-theory by the following conjecture [4, p. 550] (after slight reformulation).
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