grothendieck group over flat family

Question

Say $f:X\to Y$ is a flat family of finite length objects. If $K_0(f^{-1}(y))$ stays constant for all $y$ in an open dense set of $Y$, I believe it is then true that $K_0(f^{-1}(y))$ stays constant for all $y\in Y$. But I don't know how to prove this. Any hint on this will be appreciated.

Answer 1

Let $f\colon X=\operatorname{Spec} (\Bbb{C}[x,y]/(x^2-y)) \to Y=\operatorname{Spec} (\Bbb{C}[y])$ induced by the embedding of rings, which is flat. If $y\neq 0$ then $K_0(f^{-1}(y))=K_0(\Bbb{C}\oplus \Bbb{C})\cong \Bbb{Z} \oplus \Bbb{Z}$, and $K_0(f^{-1}(0))=K_0(\Bbb{C}[x]/(x^2))\cong \Bbb{Z}$, where the last isomorphism follows from $K_0(R)=K_0(R_{red})$ for any commutative ring $R$.