Download Cohomological invariants: exceptional groups and spin groups by Skip Garibaldi PDF

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

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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 .

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