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Lines crossing a tetrahedron and the Bloch group. (English) Zbl 1262.19003

Pragacz, Piotr (ed.), Contributions to algebraic geometry. Impanga lecture notes. Based on the Impanga conference on algebraic geometry, Banach Center, Bȩdlewo, Poland, July 4–10, 2010. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-114-9/hbk). EMS Series of Congress Reports, 297-303 (2012).
Let \(k\) be an infinite field. For \(n\geq 0\) the group \(G=\mathrm{GL}_2(k)\) acts on the free abelian group \(C(2,n)\) generated by \((n+1)\)-tuples of vectors in \(k^2\) that are pairwise linearly independent. One can form a complex \(\cdots\to C(2,2)\to C(2,1)\to C(2,0)\to0\) of \(G\)-modules and study its hyperhomology spectral sequence. By taking \(G\)-orbits of the \((n+1)\)-tuples for \(n\geq2\) one gets another complex and the ‘lines crossing a tetrahedron’ of the title become visible. As in [A. A. Suslin, Proc. Steklov Inst. Math. 183, 217–239 (1991); translation from Tr. Mat. Inst. Steklova 183, 180–199 (1990; Zbl 0741.19005)] there are connections with Milnor K-theory and the Bloch group. The main theorem describes this in terms of cross ratios.
For the entire collection see [Zbl 1245.14003].

MSC:

19D45 Higher symbols, Milnor \(K\)-theory
18G40 Spectral sequences, hypercohomology
19C20 Symbols, presentations and stability of \(K_2\)
19D55 \(K\)-theory and homology; cyclic homology and cohomology

Citations:

Zbl 0741.19005
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