By Skip Garibaldi
This quantity issues invariants of G-torsors with values in mod p Galois cohomology - within the experience of Serre's lectures within the ebook 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 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 convinced 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 varieties. This fabric on quadratic types and invariants of Spinn relies on unpublished paintings of Markus Rost. An appendix by way of Detlev Hoffmann proves a generalization of the typical Slot Theorem for 2-Pfister quadratic kinds
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This quantity issues invariants of G-torsors with values in mod p Galois cohomology - within the experience 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 hooked up E6 mod three, E7 mod three, and E8 mod five.
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Additional resources for Cohomological invariants: exceptional groups and spin groups
The argument in Step 1 of the proof of the proposition amounts to the sort of argument behind [BS, Cor. 14] combined with Sylow’s Theorem. It is essentially identical to the proof of Lemma 2 in [Ch 95], it is just written in a diﬀerent language. , at the primes 3 and 2 with H replaced by (E6 × SL3 )/µ3 and a half-spin group of type D8 respectively. This argument also appears for E6 relative to the prime 3 in [MPW, p. 153]. 9. The Rost invariant. 10) t rE 8 → H 3 (k, Q/Z(2)) H 1 (k, C × µ5 ) −−−∗−→ H 1 (k, E8 ) −−−− for every extension k/k0 .
Hence Oz has a k-point and z is in the image of the map 1 Hfppf (k, N ) → H 1 (k, G) by [DG 70, p. 373, Prop. 6b] (an fppf analogue of Prop. 37 in [Se 02]). 4. Example (char k = 0). Let G be a semisimple group. , G is of type A1 . Essentially, this is because the regular semisimple elements in V are an open subvariety. 4]. Conversely, if there is an open G-orbit in P(V ), it contains [v] for some regular semisimple element v. This v is contained in a maximal toral subalgebra of V , and by conjugacy of tori, we may assume that this subalgebra is the Lie algebra of a maximal torus T in G.
T∗ (A, α) is the neutral class in H 1 (k, F4 ), if and only if α is the reduced norm of an element of A× by [J 68, p. 416, Th. 20] or [McC, Th. 6]. [A] for uniquely determined λ1 , λ2 ∈ R3 (k0 ). But the algebra J(A, 1) is also split for every A. It follows that λ1 is zero. This proves that g3 spans Invnorm (F4 , Z/3Z). 5, we have found just three interesting invariants of F4 , namely g3 , f3 , and f5 . 7. Open problem. 4], [PeR 94, Q. 1, p. 205]) Is the map g3 × f3 × f5 : H 1 (∗, F4 ) → H 3 (∗, Z/3Z) × H 3 (∗, Z/2Z) × H 5 (∗, Z/2Z) injective?