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Projective geometry and $$K$$-theory. (English. Russian original) Zbl 0728.14008
Leningr. Math. J. 2, No. 3, 523-576 (1991); translation from Algebra Anal. 2, No. 3, 78-130 (1990).
The main aim of this paper is the construction of the motivic cohomology of line configurations in the plane. The idea of motivic cohomology goes back to Grothendieck. Then the authors apply this to some questions of projective geometry and homological algebra. For every $$n\geq 0$$ one defines the group $$A_ n(k)$$ of pairs of $$n$$-dimensional simplices in the projective space $$P^ n(k)$$ (with $$k$$ a fixed field) in analogy with the definition of the group $$A_ 2(k)$$ of pairs of triangles in the plane. Then the group $$A(k)=\oplus^{\infty}_{n=0}A_ n(k)$$ $$(A_ 0(k)={\mathbb Z}$$ and $$A_ 1(k)=$$ the multiplicative group of $$k$$) is endowed with a Hopf algebra structure. Associated with this Hopf algebra (in a standard way) is the series of complexes $$A^*(n)$$, $$n=1,2,...$$, that are components of the so-called “cobar construction”. The first nontrivial complex $$A^*(2):\;0\to A_ 2\to k^*\otimes k^*\to 0$$ is concentrated in degrees 1 and 2. One proves that $$H^ 2(A^*(2))=K_ 2(k)$$ and $$H^ 1(A^*(k))_{{\mathbb Q}}=K_ 3^{\text{ind}}(k)_{{\mathbb Q}}$$, where $$K_ n(k)$$ are the Quillen groups of $$k$$, and $$K_ 3^{\text{ind}}(k)_{{\mathbb Q}}$$ is the indecomposable part of $$K_ 3(k)$$. The basis of the construction of a motivic cohomology is a theory of mixed Hodge structures. Finally, the authors connect this theory with the so-called Bloch group.

##### MSC:
 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 19E08 $$K$$-theory of schemes