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 e-book Cohomological invariants in Galois cohomology - for numerous easy 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 provide as corollaries Pfister's effects on 10- and 12-dimensional quadratic varieties and Rost's theorem on 14-dimensional quadratic types. This fabric on quadratic varieties and invariants of Spinn is predicated on unpublished paintings of Markus Rost.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 types
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Additional info for Cohomological invariants: exceptional groups and spin groups
7], so the mod 3 portion deﬁnes a nonzero invariant of E7 with values in Z/3Z. 5. 1, we conclude that the restriction Invnorm (E7 , Z/3Z) → Invnorm (F4 , Z/3Z) k0 k0 is injective; by the above and Th. 6, it is an isomorphism. We conclude: (E7 , Z/3Z) is a free R3 (k0 )-module with basis g3 . Theorem. 3. Exercise (Mod 3 invariants of adjoint E7 ). Write E7adj for the split adjoint group of type E7 . Prove that the invariant g3 of E7 induces an invariant g3adj : H 1 (∗, E7adj ) → norm H 3 (∗, µ⊗2 (E7adj , Z/3Z) is a free R3 (k0 )-module with basis g3adj .
Exercise. , i∗ (J, β) = i∗ (J , β ) for some β, β ∈ k× , see pages 242–244 of [J 68]. , i∗ (J, 1) = i∗ (J , 1). Prove also that i∗ (J, 1) = i∗ (J, β) if and only if β is the norm of an element of J. 3) gives a functor H 1 (∗, (PGL3 ×µ3 ) × µ3 ) → H 1 (∗, E6 ) where the PGL3 ×µ3 in parentheses is the subgroup of F4 from §8. 3) after an extension of the base ﬁeld of dimension at most 2. 4) Invnorm (E6 , Z/3Z) → Invnorm (PGL3 ×µ3 × µ3 , Z/3Z) is injective. 5. An invariant of degree 3. (α) ∈ H 3 (k, µ⊗2 3 ) 32 II.
The residue of the ﬁrst term is zero and the residue of the second is a multiple of resk(π)/k a(y). The form fy is zero on the sum of the vectors in the dual basis in V ⊗ k(π), and this is a nontrivial zero because the dimension of V is not 1. It follows that ω has residue zero. This proves the claim. (fy (v)), but does not change ω0 . (fy (v)) does not depend on the choice of v. The remainder of the proposition is clear. 4. Example. The invariants produced by the proposition need not be interesting.
Cohomological invariants: exceptional groups and spin groups by Skip Garibaldi