By Skip Garibaldi
This quantity matters invariants of G-torsors with values in mod p Galois cohomology - within the feel of Serre's lectures within the booklet Cohomological invariants in Galois cohomology - for varied basic algebraic teams G and primes p. the writer determines the invariants for the phenomenal teams F4 mod three, easily attached E6 mod three, E7 mod three, and E8 mod five. He additionally determines the invariants of Spinn mod 2 for n = 12 and constructs a few invariants of Spin14. alongside the way in which, the writer proves that yes maps in nonabelian cohomology are surjective. those surjectivities supply as corollaries Pfister's effects on 10- and 12-dimensional quadratic types and Rost's theorem on 14-dimensional quadratic varieties. This fabric on quadratic kinds and invariants of Spinn is predicated on unpublished paintings of Markus Rost. An appendix by means of Detlev Hoffmann proves a generalization of the typical Slot Theorem for 2-Pfister quadratic varieties
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Extra resources for Cohomological invariants: exceptional groups and spin groups
1 We write Hfppf (k, N ) for the pointed set of k-N -torsors relative to the fppf topology as in [DG 70]. When N is smooth, this group agrees with the usual Galois cohomology set H 1 (k, N ) [DG 70, p. 6], so the reader who wishes to avoid ﬂat cohomology may simply add hypotheses that various groups are smooth or—more restrictively—only consider ﬁelds of characteristic zero. 1, Exercise 2] with B, C, D replaced by G, N, GL(V ). Proof. Write O for the G-orbit of [v] in P(V ) (equivalently, G/N ).
Note that since the L-orbit of v is dense in V , O is dense in P(V ), hence open in P(V ) because orbits are locally closed. As V is an irreducible representation of L, the torus S in the center of L acts on V by scalar multiplication. But G and S generate L, so the G- and L-orbits in P(V ) coincide. That is, the G-orbit of [v] ∈ P(V ) is open. 3 completes the proof. 11 give surjections Hfppf (K, N ) → H 1 (K, G) for every extension K/k, where N is the scheme-theoretic stabilizer of a k-point in the open 1 (∗, N ) → orbit.
1. y for a unique λ ∈ R5 (k0 ). That is, the claim holds in this case. We now consider the case where z is zero, but the extension k1 obtained by adjoining a primitive 5-th root of unity to k0 may be proper. y for a uniquely determined λ1 ∈ H 1 (k1 , µ5 ). Write λ0 for the unique class in H 1 (k0 , µ5 ) whose restriction to k1 is λ1 . 2 proves the claim. 14. CONSTRUCTION OF GROUPS OF TYPE E8 43 Finally, we consider the general case. +rE8 (η) rE8 / H 3 (k, Q/Z(2)) where τη is the twisting isomorphism.