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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.
##### MSC:
 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
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##### References:
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