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The Arason invariant and mod 2 algebraic cycles. (English) Zbl 1025.11009

From the text: Let \(k\) be a field, \(X\) over \(k\) a smooth variety with function field \(K\) and \(E\) a quadratic vector bundle over \(X\). Assuming that the generic fibre \(q\) of \(E\) is in \(I^3K\subset W(K)\), we compute the image of its Arason invariant \[ e^3(q)\in H^0(X,{\mathcal H}_{\text{ét}}^3({\mathbb Z}/2)) \] in \(\text{CH}^2(X)/2\) by the \(d_2\) differential of the Bloch-Ogus spectral sequence. This gives an obstruction to \(e^3(q)\) being a global cohomology class.
This paper is organized as follows. In Section 1 we review Arason’s invariant, and in Section 2 the special Clifford group. The heart of the paper is Sections 3 and 4, where we compute low-degree \({\mathcal K}\)-cohomology of split reductive linear algebraic groups with simply connected derived subgroups and their classifying schemes. We collect the fruits of our labor in Section 6, where we define the invariants \(\gamma_1(F)\) and \(\gamma_2 (F)\) and prove in Theorem 6.9 that \[ 2\gamma_2(F)= c_2(F)+ \gamma_1(F)^2\in \text{CH}^2(X) \] where \(c_2(F)\) is the second Chern class of the \(\text{SL}(2n)\)-tensor (a vector bundle) stemming from \(F\). Theorem 1 follows from this identity and the identification of the map \(d_2\) as a differential in a snake diagram. Theorem 1 is proven in Section 8. In Section 9 we give some applications to quadratic forms over a field.
There are 3 appendices. Appendix A shows how different models of the simplicial classifying scheme of a split torus yield the same \({\mathcal K}\)-cohomology. Appendix B presents a construction and a characterization of the invariant defined by Serre and Rost for torsors under a simple, simply connected algebraic group \(H\) over a field: in the case of Spin, this allows this paper to be self-contained. Let us point out that our method tackles the \(p\)-primary part of the Rost invariant as well, in case \(\operatorname{char} k=p> 0\). Finally, Appendix C compares \({\mathcal K}\)-cohomology of the simplicial scheme \(BH\) with that of an approximating variety \(B_r H\): it turns out that they do not coincide. In this last appendix, we have to stay away from the characteristic of \(k\) if it is nonzero.
The group \(H^1(G,{\mathcal K}_2)\) was first computed by P. Deligne at the end of the seventies for any \(G\), semisimple, simply connected, and not necessarily split. Our method here is different from this.

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
14C25 Algebraic cycles
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
11E72 Galois cohomology of linear algebraic groups
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