By Kahane J.-P.
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Additional info for Two problems on trigonometric series
Example text
By convention Ztr ((X, x)∧0 ) = Z and Ztr ((X, x)∧q ) = 0 when q < 0. 13. The presheaf Ztr ((X1 , x1 ) ∧ · · · ∧ (Xn , xn )) is a direct summand of Ztr (X1 × · · · × Xn ). In particular, it is a projective object of PST. Moreover, the following sequence of presheaves with transfers is split-exact: {xi } 0 → Z → ⊕i Ztr (Xi ) → ⊕i, j Ztr (Xi × X j ) → · · · · · · → ⊕i, j Ztr (X1 × · · · Xˆi · · · Xˆ j · · · × Xn ) → ⊕i Ztr (X × · · · Xˆi · · · × Xn ) → → Ztr (X1 × · · · × Xn ) → Ztr (X1 ∧ · · · ∧ Xn ) → 0.
11). 2. For any scheme T over k, Ztr (T ) is an e´ tale sheaf. P ROOF. Since PST(k) is an additive category, we have the required decomposition of Ztr (T )(X Y ) = HomCork (X Y, T ). 2. As U × T → X × T is flat, the pullback of cycles is well-defined and is an injection. Hence the subgroup Ztr (T )(X) = Cork (X, T ) of cycles on X × T injects into the subgroup Ztr (T )(U) = Cork (U, T ) of cycles on U × T . 1 (1) is exact at Ztr (T )(U), take ZU in Cork (U, T ) whose images in Cork (U ×X U, T ) coincide.
It will consist of three steps: (1) Construction of θ : H n,n (Spec F, Z) → KnM (F). 5. (2) Construction of λF : KnM (F) → H n,n (Spec F, Z). 6. 8. 9) ⇒ ∃ λF (3) Proof that these two maps are inverse to each other. 11). Before starting the proof of the theorem we need some additional properties of motivic cohomology and Milnor K-theory. 10 is the group of zero cycles of (A − 0) . 2. We have H p,q (Spec F, Z) = Hq−p C∗ Ztr (G∧q m )(Spec F) for all p and q. In particular we have H n,n (Spec F, Z) =H0 C∗ Ztr (G∧n m )(Spec F) 1 = coker Ztr (G∧n m )(A ) ∂0 − ∂1 ✲ Ztr (G∧n )(Spec F) .