By Marc Levine
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Extra resources for A short course in K-theory
10 (Beilinson, Soul´e). Let F be a field. Then Kp (F )(q) = 0 if p > 0, 2q ≤ p. As mentioned above, Soul´e showed that Kp (F )(1) = 0 for p > 1, which verifies the conjecture for p ≤ 3. Except for number fields, finite fields and function fields of curves over finite field, and some trivial extensions of these examples, the conjecture is unknown in the first interesting cases n = 4, 5, that is, the weight two part is not known to vanish. The Beilinson-Soul´e vanishing conjecture is in turn related to other conjectures of Bloch and Beilinson concerning the existence of a category of mixed motives with certain properties.
Since the Quillen spectral sequence is functorial for flat morphisms, we may restrict to the open neighborhood X \ D of Z to compute the composition dq−1,−1 i pz w 1 k(W )× −→ ⊕x∈X (q−1) k(x)× −− −−→ ⊕x∈X (q) Z − → Z, where pz is the projection on the summand indexed by the generic point z of Z. Since the term k(w)∗ clearly goes to zero upon this restriction, it follows that pz ◦ d1 ◦ iw (f ) = 0. Now suppose that Z ⊂ D. We may again use functoriality of the localization sequence with respect to finite morphisms to reduce to the case X = D.
The stalk Kp,x at x ∈ X is just Kp (OX,x ). 3) 0 → Kq → ⊕x∈X (0) ix∗ (Kq (k(x))) − → d d 1 1 → ⊕x∈X (q) ix∗ (K0 (k(x))). ⊕x∈X (1) ix∗ (Kq−1 (k(x))) − → ... − Here ix : x → X is the inclusion, and we consider Kn (k(x)) as the constant sheaf on the one-point space x. 3) is exact. 3) gives a flasque resolution of the sheaf Kq . 3. Suppose Gersten’s conjecture is true for all the local rings OX,x . 2). In particular, H p (X, Kq ) = 0 if p > q. 3) (after deleting Kq ). 4. Take the case q = 1. Then K1 is just the sheaf of units × OX , and the sheafified Gersten complex is (assume X is irreducible with generic point η) × 1 → OX → iη∗ k(X)× → ⊕x∈X (1) ix∗ Z → 0.