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The \(K\)-theory of versal flags and cohomological invariants of degree 3. (English) Zbl 1386.19001

Let \(G\) be a split semisimple linear algebraic group over a field \(F\). Let \(U'\) be the versal \(G\)-torsor in the sense of S. Garibaldi et al. [Cohomological invariants in Galois cohomology. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1159.12311)]. The versal flag is \(X:=U'/B\) where \(B\) is a Borel subgroup of \(G\). The authors compute the Grothendieck ring \(K_0(X)\) when \(G=G^{sc}/\mu_2\), where \(G^{sc}\) is simply connected of Dynkin type A or C. This involves solving a system of linear equations over a Laurent polynomial ring \(R[x_1^{\pm 1},\dots,x_n^{\pm1}]\), where \(R\) equals \(\mathbb Z\) or \({\mathbb Z}/m{\mathbb Z}\), \(m\geq2\). They show the system has ‘flatness’ properties that make it amenable to solving.
Next they compute various groups of cohomological invariants of degree 3 when \(G\) is of the form \((H_1\times H_2)/\mu_k\) where \(H_1\), \(H_2\) are simply connected and have the same Dynkin type.

MSC:

19-XX \(K\)-theory
14M17 Homogeneous spaces and generalizations
20G15 Linear algebraic groups over arbitrary fields
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry

Citations:

Zbl 1159.12311
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References:

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