Rudiments of Algebraic Geometry by W.E. Jenner

By W.E. Jenner

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6 is proven. Let us finish this section with the definition of the higher Witt indices and the splitting pattern of a quadric. Since this notion plays an important role throughout the paper, I should emphasize that the definition of splitting pattern I use somewhat deviates from the common usage. To make it explicit, let k be a field of characteristic different from 2 and let q a quadratic form defined over k. We construct a sequence of fields and quadratic forms in the following way. Set k0 := k, i0 (q) := iW (q), the Witt index of q, and q0 := qan , the anisotropic kernel of q.

Si x est s´eparable, l’hypoth`ese sur k implique que E = k(x) est cyclique sur k. Soit g un g´en´erateur de Gal(E/k). Par le th´eor`eme 90 de Hilbert, on peut ´ecrire x = gy/y pour un y ∈ E ∗ convenable. Par le th´eor`eme de Skolem–Noether, g se prolonge en un automorphisme int´erieur de A, donc x est un commutateur dans A∗ . 2. Pour toute alg`ebre centrale simple A d’indice e, on a Éepi SK1 (A) = 0, o`u les pi d´ecrivent l’ensemble des facteurs premiers de e. Preuve. On se r´eduit encore au cas o` u A est un corps, e est une puissance d’un nombre premier p et toute extension finie de k est de degr´e une puissance de p.

At the same time, we have some results which guarantee that particular elements of Λ(Q) are not connected. 2 shows that the Tate motives Z, Z(1)[2], . . , Z(i1 (q)−1)[2i1 (q)−2] all belong to different connected components of Λ(Q). Here is a generalization of this result. 1 The (incremental) splitting pattern of a quadratic form or a quadric is defined at the end of Sect. 13 ([26, Corollary 2]). Let Q be a smooth projective quadric, and N be an indecomposable direct summand of M (Q) such that iW (q|Ft ) ≤ a(N ) < iW (q|Ft+1 ).

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