Constructing analog of loop space of algebraic $K$-theory at the level of varieties

Question

I have a somewhat open-ended and vague question regarding algebraic $K$-groups. According to the fundamental theorem of algebraic $K$-theory for a regular and Noetherian ring $R$, we have $K_i(R[x,x^{-1}])\cong K_i(R)\oplus K_{i-1}(R)$. The interesting fact about this for me is, for each variety, we can find a variety that its $i$-th $K$-group contains the information of the $i-1$-th $K$-group of our initial variety. My question is the reverse of this. For each regular variety $X$, is it possible to find a regular variety $Y$ such that $K_i(Y)$ contains $K_{i+1}(X)$ as a direct summand? The closest for this that I've find out is the following. If $N$ is the affine nodal curve, $KH_i(X\times N)\cong KH_i(X)\oplus KH_{i+1}(X)$. Of course here $N$ is not regular and $KH$ is the homotopy $K$-theory.

Answer 1

I don't think so.

If such a $Y$ exists, $K_{-1}(Y)$ contains $K_0(X)$ as a direct summand. If $K$ is Quillen K-theory (aka connective K-theory), $K_{-1}(Y)=0$ anyway, and even if it is Thomason-Trobaugh K-theory (aka non-connective K-theory), $K_{-1}(Y)=0$ because you also demand that $Y$ be regular. Hence, this would force $K_0(X)$ zero, which is impossible for a nonempty variety (rationalized $K_0$ has rationalized $CH_0$ as a direct summand, and the latter has the nontrivial degree map to $\mathbb{Q}$. It must be nonzero because a single closed point will be sent to its residue field degree under this map).

So, with either interpetation of your K-theory, there is a problem around $K_0$. Of course, your property might still hold in higher degrees, but then it's certainly not a loop space in a conventional sense.