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How to compute the integer corresponding to a class in $G_0(B_{\mathrm{red}})$ for a commutative noetherian ring $B$?

$\newcommand{\red}{\mathrm{red}}$Let $k$ be an algebraically closed field of characteristic zero and $m$ be a positive integer. Let $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,...