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Cohomological invariants associated to symmetric bilinear forms. (English) Zbl 0761.12003

This exposition describes how to apply the machinery of the author’s homotopy category of simplicial sheaves to reprove a number of results relating algebraic and topological invariants of Galois representations.
Let \(K\) be a field of characteristic different from \(\mathbb{Q}\) and let \(\rho:\Omega_ K\to O_ n(K)\) be an orthogonal Galois representation. Topologically \(\rho\) has Stiefel-Whitney classes, \(SW_ i(\rho)\in H^ i(K;\mathbb{Z}/2)\) obtained by pulling back a universal class, \(w_ i\in H^ i(BO(K^{sep});\mathbb{Z}/2)\). In addition, the spinor norm, \(Sp:O_ n(K)\to H^ 1(K;\mathbb{Z}/2)\) defines a class, \(Sp_ 2[\rho]\in H^ 2(K;\mathbb{Z}/2)\). Finally, by descent theory, \(\rho\) can be considered as a symmetric bilinear form which has classes \(HW_ i(\rho)\in H^ i(K;\mathbb{Z}/2)\) generalizing the Hasse invariant, \(HW_ 2(\rho)\).
Relationships between these classes, in dimension 2, were discovered by J.-P. Serre when \(\rho\) is a permutation representation. This fundamental discovery was elaborated upon by A. Fröhlich (in dimension 2), B. Kahn and the reviewer (independently; in higher dimensions). The results are collected in [V. P. Snaith, Topological methods in Galois representation theory, Ch. III–IV (1989; Zbl 0673.12009)]. In the homotopy category of simplicial sheaves on the etale site, \((Sh|_ K)_{et}\), \(\rho\) defines a class in \([*,BO_ n]\) where \(n=\text{rank}(\rho)\). The author calculates \(H^*(BO_ n;\mathbb{Z}/2)\) in this homotopy category, obtaining a polynomial ring on Stiefel-Whitney classes. It transpires that \(BO_ n\) is homotopy equivalent to \(BO_ \rho\), the classifying object corresponding to \(\rho\) as a bilinear form. Hence \(H^*(BO_ \rho;\mathbb{Z}/2)\) is also a polynomial algebra and the comparison of the two sets of polynomial generators yields the relations between cohomology classes which was referred to above.
The proofs, although in a more abstract setting than they may be used to, will have a familiar feel to the experts. This abstraction is vindicated in a forthcoming paper in which the author generalizes \(Sp_ 2(\rho)\) to arbitrary dimensions and obtains the general formula relating all the \(HW_ i(\rho)\), \(Sp_ i(\rho)\) and \(SW_ i(\rho)\).

MSC:

12G05 Galois cohomology
55P42 Stable homotopy theory, spectra
11E72 Galois cohomology of linear algebraic groups
11R34 Galois cohomology
11R32 Galois theory

Citations:

Zbl 0673.12009
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