Suppose $X$ is a smooth, projective variety. Then Bloch's formula states that $$CH^n(X)\cong H^n(X, \mathcal{K}_{n,X})$$
where $\mathcal{K}_{n,X}$ is the sheaf associated to the presheaf $U=Spec{A} \mapsto K_n(A)$, the $n$-th K group. An easy way to understand this is via the Gersten resolution of the sheaf $\mathcal{K}_{n,X}$.
My question is the following: If $Y$ is a non-reduced scheme such that $Y_{red}=X$, then what do we know about the relation between $H^2(Y,\mathcal{K}_{2,Y})$ and $CH^2(Y)$?
Thanks in advance.
