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The group \(SK_ 2\) for quaternion algebras. (Russian) Zbl 0652.12006

The starting point of this paper is a problem from the “Algebraic \(K\)-theory” of H. Bass [New York etc.: W. A. Benjamin (1968; Zbl 0174.30302)]: If \(A\) is a semisimple algebra over the field \(F\) is always \(SK_1(A)=0\)? V. P. Platonov constructed an example for which \(SK_1(A)\ne 0\) [Izv. Akad. Nauk SSSR, Ser. Mat. 40, 227–261 (1976; Zbl 0338.16005)].
In this paper there are considered a field \(F\) of characteristic different from \(2\) and the quaternionic \(F\)-algebra \(D\) generated by two elements \(a, b\) from \(F^*\). There are studied the groups \(K_2(D)\), \(K_2(F)\) and the reduced norm morphism \(\text{Nrd}\) from \(K_2(D)\) to \(K_2(F)\). In the main result of the paper it is proved that \(\text{Nrd}\) is injective (i.e. \(SK_2(D)=0)\). There is also described the image of \(K_2(D)\) in \(K_2(F)\). These results are obtained from the consideration of the category \(M(X,D)\) of the coherent \(\mathcal O_X\)-modules with the operators \(D\) (\(X\) being an \(F\)-variety) and the consideration of the \(K^D\)-theory of quadrics in a projective space. Many of these results are interesting in themselves. We mention the study of the group of classes of \(K^D\)-cycles.

MSC:

11R70 \(K\)-theory of global fields
11R52 Quaternion and other division algebras: arithmetic, zeta functions
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
16P10 Finite rings and finite-dimensional associative algebras
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19D45 Higher symbols, Milnor \(K\)-theory
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