By Marc Levine
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Additional resources for A short course in K-theory
1) breaking up into short exact sequences jp∗ p/p+1 0 → Kq (MpX ) − → Kq (MX ∂p ) −→ Kq−1 (Mp+1 X ) → 0 for all p and q. 3. Cohomology of K-sheaves. One can go even further with this if one considers the K-sheaves Kp on X, defined as the sheaf associated to the presheaf U → Kp (U ). 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.
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.
Op´erations en K-th´eorie alg´ebrique. Canad. J. Math. 37 (1985), no. 3, 488–550. , Torsion 0-cycles on affine varieties in characteristic p, J. Algebra 120 (1989), no. 2, 428–432.  Suslin, A. Algebraic K-theory of fields. Proceedings of the International Congress of Mathematicians, Vol. , 1986), 222–244, Amer. Math. , Providence, RI, 1987. , 1046, Springer, Berlin, 1984.  Suslin, A. and Voevodsky, V. M. Annals of Mathematics Studies, 143, Princeton University Press, Princeton, NJ, 2000.