Baek, Sanghoon; Devyatov, Rostislav; Zainoulline, Kirill The \(K\)-theory of versal flags and cohomological invariants of degree 3. (English) Zbl 1386.19001 Doc. Math. 22, 1117-1148 (2017). 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. Reviewer: Wilberd van der Kallen (Utrecht) Cited in 2 Documents 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 Keywords:linear algebraic group; twisted flag variety; ideal of invariants; versal torsor; cohomological invariant Citations:Zbl 1159.12311 PDFBibTeX XMLCite \textit{S. Baek} et al., Doc. Math. 22, 1117--1148 (2017; Zbl 1386.19001) Full Text: DOI arXiv References: [1] S. Baek, Chow groups of products of Severi-Brauer varieties and invariants of degree 3, Trans. Amer. Math. Soc. 369(3): 1757-1771, 2017. · Zbl 1391.14010 [2] H. Bermudez, A. Ruozzi, Degree 3 cohomological invariants of split simple groups that are neither simply connected nor adjoint, J. Ramanujan Math. Soc. 29(4): 465-481, 2014. · Zbl 1328.20063 [3] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics , 150. Springer-Verlag, New York, 1995. xvi+785 pp. [4] S. Garibaldi, A. Merkurjev, J.-P. Serre, Cohomological Invariants in Galois Cohomology, University Lecture Series 28, AMS, Providence, RI, 2003. · Zbl 1159.12311 [5] S. Garibaldi, K. Zainoulline, The gamma-filtration and the Rost invariant, J. Reine und Angew. Math. 696: 225-244, 2014. · Zbl 1345.14022 [6] S. Gille, K. Zainoulline, Equivariant pretheories and invariants of torsors, Transf. Groups 17(2): 471-498, 2012. · Zbl 1291.14012 [7] N. Karpenko, Chow ring of generic flag varieties, to appear in Math. Nachr. · Zbl 1378.14008 [8] N. Karpenko, Chow ring of generically twisted varieties of complete flags, Adv. Math. 306: 789-806, 2017. · Zbl 1466.14009 [9] N. Karpenko, Chow groups of some generically twisted flag varieties, Ann. K-Theory 2(2): 341-356, 2017. · Zbl 1362.14006 [10] N. Karpenko, On topological filtration for Severi-Brauer varieties, Proc. Sympos. Pure Math. 58: 275-277, 1995 · Zbl 0832.19003 [11] Y. Laszlo, C. Sorger, The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. Ec. Norm. Sup\'er. (4) 30(4): 499-525, 1997. · Zbl 0918.14004 [12] A. Merkurjev, Degree three unramified cohomology of adjoint semisimple groups, Preprint 2016. · Zbl 1372.12002 [13] A. Merkurjev, Cohomological invariants of central simple algebras, Izvestia RAN, Ser. Mat., 80(5): 869-883, 2016. · Zbl 1353.16007 [14] A. Merkurjev, Degree three cohomological invariants of semisimple groups, J. Eur. Math. Soc. 18(2): 657-680, 2016. · Zbl 1367.12003 [15] A. Merkurjev, A. Neshitov, K. Zainoulline, Cohomological invariants in degree 3 and torsion in the Chow group of a versal flag, Compositio Math. 151(8): 1416-1432, 2015. · Zbl 1329.14017 [16] T. Mora, M. Sala, On the Gr\"obner bases of some symmetric systems and their applications to coding theory, J. Symbolic Computation 35: 177-194, 2003. · Zbl 1042.13017 [17] I. Panin. On the algebraic K -theory of twisted flag varieties. K-Theory 8(6): 541-585, 1994. · Zbl 0854.19002 [18] B. Totaro, The Chow ring of a classifying space. Algebraic K-theory (Seattle, WA, 1997), 249-281, Proc. Sympos. Pure Math. 67, AMS, Providence, RI, 1999. · Zbl 0967.14005 [19] K. Zainoulline, Twisted gamma-filtration of a linear algebraic group, Compositio Math. 148(5): 1645-1654, 2012. · Zbl 1279.14011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.