All Questions

For regular schemes $X$ and $Y$, and a faithfully flat morphism $f:Y \to X$, there is a flat pullback map of Grothendieck groups: $$ f^*:K^0(X) \to K^0(Y). $$ Is this map injective?

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence $$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to \...