Let $R$ be a domain, like $\mathbb{Z}$, for example, and let $R\to R[x]/I$ be an integral extension, like $\mathbb{Z}[i]$, for example.

Is anything known about the relative algebraic $K$ groups, $K_*(R',R)$? For example, when $R\to R'$ is a nilpotent extension, the Dennis trace map from relative $K$ theory to relative negative cyclic homology is an isomorphism.

Is there a similar "simpler" spectrum that computes the relative $K$-theory of a quadratic extension?

Let $R$ be a domain, e.g. $\mathbb{Z}$, and let $R\to R[x]/I$ be an integral extension, e.g $\mathbb{Z}[i]$.

Is anything known about the relative algebraic $K$ groups, $K_*(R',R)$? For example, when $R\to R'$ is a nilpotent extension, the Dennis trace map from relative $K$ theory to relative negative cyclic homology is an isomorphism.

Is there a similar "simpler" spectrum that computes the relative $K$-theory of a quadratic extension?