On the algebraic K–theory of double points

In this paper, we use trace methods to study the algebraic $K$-theory of rings form $R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2$. We compute relative $p$-adic $K$ groups for $R$ a perfectoid ring. particular, get integral when is finite field, and $K_*(R[x_1,\ldots, x_d)^2, (x_1,\ldots, x_d))$ perfect $\mathbb{F}_p$-algebra. conclude paper with some other notable computations, including which are not quite above form.