By Skip Garibaldi
This quantity issues invariants of G-torsors with values in mod p Galois cohomology - within the feel of Serre's lectures within the publication Cohomological invariants in Galois cohomology - for varied uncomplicated algebraic teams G and primes p. the writer determines the invariants for the outstanding teams F4 mod three, easily hooked up 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 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 This fabric on quadratic kinds and invariants of Spinn is predicated on unpublished paintings of Markus Rost.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 types
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Additional info for Cohomological invariants: exceptional groups and spin groups
Whenever J is reduced), so Prop. (β) is a well-deﬁned invariant of E6 . 8). 12. 9. Proposition. Invnorm (E6 , Z/3Z) is a free R3 (k0 )-module with basis g3 , g4 . k0 Proof. 6. The restriction map i∗ : Invnorm (E6 , Z/3Z) → Invnorm (F4 × µ3 , Z/3Z) k0 k0 is an injection. The center of E6 is contained in a maximal split torus, hence the image of the map H 1 (∗, µ3 ) → H 1 (∗, E6 ) is zero. 6, we (E6 , Z/3Z). 10. Exercise (Mod 3 invariants of the quasi-split E6 ). For K a quadratic ﬁeld extension of k0 , write E6K for the simply connected quasi-split group of type E6 associated with the extension K/k.
Then T ﬁxes v, so it is contained in the stabilizer N of [v]. We have: 1 ≤ rank G ≤ dim N = dim G − dim P(V ) = 1. 5. A non-example is furnished by a representation V of G on which G acts trivially. If dim V = 1, then P(V ) is a point, N equals G, and the conclusion of Th. 3 is uninteresting. If dim V is at least 2, then P(V ) does not have an open orbit (exercise). 6. Example (Reducible representations). 3, and suppose that there is a proper G-invariant subspace W of V . The quotient map V → V /W gives a G-equivariant rational surjection f : P(V ) P(V /W ).
517]. This isomorphism identiﬁes Invnorm (Alb, Z/3Z) with Invnorm (F4 , Z/3Z). 2. ig (c)), where ig is the inner autormorphism of M deﬁned by g. Since (g, z) ﬁxes both 1 and N , it belongs to the group F4 . 3) t∗ : H 1 (∗, PGL3 ) × H 1 (∗, µ ) → H 1 (∗, F4 ) ∼ = Alb(∗). 3 The image of a pair (A, α) is often denoted by J(A, α); such algebras are known as ﬁrst Tits constructions, cf. A]. 3. 5. Invariants of F4 mod 3. (α) for δ as deﬁned in §6. We now give two arguments that g3 is the restriction of an invariant of F4 .