Bottom supported Bloch groups on a codimension two curve

The ambient $U\times {\mathrm{\Delta }}^n$ has pure dimension $3+n$ and $C\times {\mathrm{\Delta }}^n$ has dimension $1+n$. Every integral closed subscheme contained in the latter therefore has codimension at least two in its ambient component. Thus none is a generator of $z_C^j(U,n)$ for $j=0,1$, proving the first assertion in every degree.

In degree zero, an integral codimension-two subscheme contained in $C$ must have dimension one, hence equals $C$ with its reduced structure. Thus $z_C^2(U,0)=\mathbb{Z}[C]$. In degree one the generators have dimension two in $U\times {\mathbb{A}}^1$ and are contained in the integral two-dimensional scheme $C\times {\mathbb{A}}^1$. The only such integral closed subscheme is the whole product, which is admissible. At each endpoint its generic local quotient length is one, so its boundary is $[C]-[C]=0$, with either overall orientation.

The complex is concentrated in nonnegative homological degrees, so its zeroth homology is its degree-zero group modulo the image of the degree-one boundary. Step 1.2 makes that image zero. On an open containing $\eta$ the same calculation applies to the integral curve $C\cap V$ and restriction takes $[C]$ to $[C\cap V]$ without changing its coefficient. Hence it is the asserted isomorphism.