$\DeclareMathOperator\Pic{Pic}$For a commutative ring with unity $R$, if $R_{\mathrm{red}}$ is semi-normal then we have an equivalent criterion which states that $\Pic(R) \cong \Pic(R[t])$. Here $\Pic(R)$ is the Picard group of $R$. So naturally it is well understood that $\Pic(R) \ncong \Pic(R[s])$ when $R = k[t^2,t^3]$ a cusp, which is a domain that is not semi-normal.

With the help of conductor ideals I was able to show that $\Pic(R) \cong k$ but I am unable to explicitly compute $\Pic(R[s])$ will the same approach via conductor ideals work, for any suggestion or guidance I am thankful.