Rost nilpotence for twisted Milnor hypersurfaces

𝑛-2 𝑖=0 M(P 𝑛 𝑘 )(𝑖) ⊕ M(Spec 𝐿) (𝑛 -1).(0.1) Assume now that 𝜑 is of the form 𝜑 ′ ⊕ 𝑑𝑖𝑎𝑔(𝑎 𝑚+1 , . . ., 𝑎 𝑛 ), 𝑛 > 𝑚 ≥ 2, where 𝑎 𝑖 ∈ 𝑘 × , and 𝐿 ′ = 𝑘 [𝜑 ′ ] is a proper Galois algebra of degree 𝑚 + 1 in the sense of [KMRT, Ch.V.18B] that is a finite étale algebra with suitable group action.So 𝐿 is the product of 𝐿 ′ with 𝑛 -𝑚 copies of 𝑘.Then the computation of M(𝑌 ) of the respective hyperplane section can be reduced to that of its Galois part M(𝑌 ′ ) using the following analogue of the Rost filtration for quadrics: