I am reading two papers by Daniel Grayson: "Localization for flat modules in algebraic K-theory" and "Algebraic cycles and algebraic K-theory" and I am wondering if any recent advances in K-theory can extend the results a bit. Specifically, suppose $\pi: X \to S$ is a smooth projective morphism of relative dimension 2. Mimicking Corollary 1.6 in the second paper, can we write an exact complex of sheaves $$ 0 \to \mathcal{K}_3 (X) \to \mathcal{K}_3^{0/1}(X/S) \to \mathcal{K}_2^{1/2}(X/S) \to \mathcal{K}_1^2(X/S) \to 0? $$ The sheaves involved are sheafification of K-theory in Zariski or etale topology. The last term is K-theory of coherent sheaves flat over $S$ with support of relative codimension 2 (i.e. support finite over $S$). The two intermediate terms are some kind of relative K-theory for the pairs built from coherent sheaves (flat over $S$) with support of relative codimension 0 (resp. 1) and its subcategory of sheaves with support of relative codimension 1 (resp. 2).
In fact, I am OK with a resolution of this nature where the two middle terms can be a bit more obscure than stated, as long as the first and the last terms remain the same.
An alternative: when $Y \to S$ is a flat family of projective curves, but not necessarily smooth, is there an analogue of the resolution $$ 0 \to \mathcal{K}_2(Y) \to \mathcal{K}_2^{0/1}(Y/S) \to \mathcal{K}_1^1(Y/S) \to 0 $$ stated in Corollary 1.6 for the case when $Y$ is relatively smooth over $S$? Again, it's OK if the middle term gets a bit more obscure, as long as the other two survive (maybe with the correction that the first term is K-theory of coherent sheaves, not of projective modules).
