Note on the cohomological invariant of Pfister forms (Q2839627)

Let \(G\) be an algebraic group over a field \(k\) of characteristic \(\text{ch}(k) \neq 2\). The cohomological invariant ring \(\text{Inv}^ {\ast }(G; \mathbb Z/2\mathbb Z) = \oplus _ i \text{Inv} ^ i (\mathbb Z/2\mathbb Z)\) of \(G\) is the ring generated by natural functors \(H ^ 1(K; G) \to H ^ i(K; \mathbb Z/2\mathbb Z)\), for the category of finitely-generated fields \(K\) over \(k\) (see [\textit{S. Garibaldi} et al., Cohomological invariants in Galois cohomology. Providence, RI: AMS (2003; Zbl 1159.12311)]). Moreover, one can define the cohomological invariant ring \(\text{Inv}^ {\ast }(\text{Pfister}_ n; \mathbb Z/2\mathbb Z)\) of \(n\)-Pfister forms, although the corresponding group \(G\) does not exist, for \(n \geq 4\). This ring has been computed by Serre, as noted by the authors, by elementary but very elegant arguments (see Theorem 18.1, loc. cit.). The paper under review shows that \(\text{Inv}^ {\ast }(\text{Pfister}_ n; \mathbb Z/2\mathbb Z)\) is isomorphic to \(\text{Inv}^ {\ast }((\mathbb Z/2\mathbb Z) ^ n; \mathbb Z/2\mathbb Z) ^ {\text{GL}_ n(\mathbb Z/2\mathbb Z)}\), the invariant subring of \(\text{Inv}^ {\ast }((\mathbb Z/2\mathbb Z) ^ n; \mathbb Z/2\mathbb Z)\) under the action of the linear group \(\text{GL}_ 2(\mathbb Z/2\mathbb Z)\) on the elementary abelian group \((\mathbb Z/2\mathbb Z) ^ n\) of rank \(n\). The proof relies on machinery from motivic cohomology and Dickson algebras. Specifically, it depends on the Milnor conjecture and the Beilinson-Lichtenbaum conjecture, both proved by \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 98, 59--104 (2003; Zbl 1057.14028)]; Ann. Math. (2) 174, No. 1, 401--438 (2011; Zbl 1236.14026)].