Premium
Motivic Cohomology of Pairs of Simplices
Author(s) -
Zhao Jianqiang
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014709
Subject(s) - mathematics , coproduct , cohomology , motivic cohomology , sheaf , pure mathematics , betti number , simple (philosophy) , dimension (graph theory) , hopf algebra , de rham cohomology , combinatorics , algebra over a field , discrete mathematics , equivariant cohomology , philosophy , epistemology
For an arbitrary field F we study the double scissors congruence groups A n ( F ) generated by admissible pairs of simplices in the projective n ‐spaceP F n over F . Let ( L , M ) be an admissible pair of simplices whose faces are defined as hyperplanes inP F n such that they do not have common faces of the same dimension. When L and M are in general positions we define a linearly constructible motivic perverse sheaf producing a motivic cohomology whose Betti realization is the relative cohomology H ∗ (P F n ∖ L , M ∖ L ; Q ) . The motivic cohomology provides a simple explanation of why the generic part of A • forms a Hopf algebra with well‐defined coproduct. When n = 2 we define explicitly the coproduct on A 2 which simplifies the approach of Beilinson et al. We also complete the same task for A 3 and A 4 which enables us to develop further results in another paper connecting Aomoto trilogarithms to the classical ones. 2000 Mathematics Subject Classification 14F42, 16W30 (primary), 13D03, 13D25 (secondary).