On the First Witt Index of Quadratic Forms

We prove Hoffmann’s conjecture determining the possible values of the first Witt index of anisotropic quadratic forms of any given dimension. The proof makes use of the Steenrod type operations on the modulo 2 Chow groups constructed by P. Brosnan. Let F be a field of characteristic 6= 2. For an anisotropic quadratic form φ over F with dim(φ) ≥ 2, the first Witt index i1(φ) is the Witt index (i.e., the dimension of a maximal totally isotropic subspace) of the form φF (φ) over the function field F (φ) = F (Xφ) of the projective quadric given by φ. Clearly, this i1(φ) is the minimal positive Witt index of φE, when E runs over all field extension of F . We are going to prove the following Conjecture 0.1 (Hoffmann). Let us write the integer dimφ − 1 as a sum of powers of 2: dim(φ)− 1 = 21 + 22 + · · ·+ 2r with 0 ≤ n1 < n2 < · · · < nr. Then the integer i1(φ) − 1 is a partial sum of this sum (including the empty one and not including the whole sum): i1(φ)− 1 = 2 n1 + 22 + · · ·+ 2s for some 0 ≤ s ≤ r − 1. We remark that for any given n ≥ 2, all the values of i1(φ), prescribed by Conjecture 0.1 for forms φ with dim(φ) = n, are really possible, that is, do occur for suitable φ over suitable F . To see it, we take a field k with an anisotropic r-fold Pfister form 〈〈a1, . . . , ar〉〉. Then we take indeterminates t0, . . . , tm, set F = k(t0, . . . , tm), and notice that the first index of the quadratic F -form q = 〈〈a1, . . . , ar〉〉⊗ 〈t0, . . . , tm〉 is equal to 2 . Therefore, by [4, lemma 7.3], for every j = 0, 1, . . . , 2 − 1, the first Witt index of certain j-codimensional subform φ ⊂ qF̃ over certain purely transcendental field extension F̃ /F is 2 r − j (while dim(φ)− i1(φ) is still 2 m). 1 We prove Conjecture 0.1 in §3. In §1 and §2 we introduce two different tools needed in the proof. Date: 14 June 2002.