On the algebraizability of formal deformations in $K$-cohomology

We show that algebraizability of the functors $R^1{\pi }_{\ast }{\mathcal{K}}_{2,X}^M$ and $R^2{\pi }_{\ast }{\mathcal{K}}_{2,X}^M$ is a stable birational invariant for smooth and proper varieties $\pi :X\to k$ defined over an algebraic extension $k$ of $\mathbb{Q}$. The same is true for the étale sheafifications of these functors as well. To get these results we introduce a notion of relative $K$-homology for schemes of finite type over a finite dimensional, Noetherian, excellent base scheme over a field. We include this material in an appendix.