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Dilogarithm, Grassmannian complex and scissors congruence groups of algebraic polyhedra. (English) Zbl 0849.14003
Fix an infinite field \(k\). A. Beilinson, R. MacPherson and V. Schechtman [Duke Math. J. 54, 679-710 (1987; Zbl 0632.14010)] defined the Grassmannian complex of weight \(p\), \(G(p)\), and indicated its relation with higher polylogarithms. In particular, \(G(2)\) determines the Grassmannian dilogarithm. In: The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 135-172 (1990; Zbl 0737.14003), A. A. Beilinson, A. B. Goncharov, V. V. Schechtman and A. N. Varchenko introduced groups \(A_n\) of pairs of triangles (modulo scissors congruence) in \(\mathbb{P}^n\) and related complexes \(A(n)\).
For \(n = 2\) one has \(A(2) : A_2 @>\nu>> A_1 \otimes A_1 \cong k^*_\mathbb{Q} \otimes k^*_\mathbb{Q}\), and the cohomology of \(A(2)\) is related to algebraic \(K\)-groups as follows: \(H^1 (A(2)) \cong K_3^{\text{ind}} (k)_\mathbb{Q}\) (the indecomposable part of \(K_3 (k))\) and \(H^2 (A(2)) \cong K^M_2 (k)_\mathbb{Q} \) (the Milnor \(K\)-group). Here a complex \(T(2) : T_2 (2) @>\partial>> T_1(2)\) is defined. \(T_2 (2)\) is the quotient \(\mathbb{Q}\)-vector space of the \(\mathbb{Q}\)-vector space freely generated by so-called admissible triangles in \(\mathbb{P}^2_k\) by a set of relations including the scissors congruences. \(T_1(2)\) is defined as the quotient of the \(\mathbb{Q}\)-vector space of generic line elements in \(\mathbb{P}^2_k\) by a set of relations also including the scissors congruences. On the one hand one has the result that the complexes \(T(2)\) and \(A(2)\) are isomorphic. This is proved by explicit calculation. It may sometimes be advantageous to work with \(T(2)\) instead of \(A(2)\). On the other hand two maps of complexes \(u,f : G(2) \to T(2)\) are constructed. These maps are related to Grassmannian dilogarithms.
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
51M20 Polyhedra and polytopes; regular figures, division of spaces
14A20 Generalizations (algebraic spaces, stacks)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
19D45 Higher symbols, Milnor \(K\)-theory
19D55 \(K\)-theory and homology; cyclic homology and cohomology
Full Text: Numdam EuDML
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