Relative big polynomial rings

Abstract: Let $K$ be the field of Laurent series with complex coefficients, let $\mathcal{R}$ be the inverse limit of the standard-graded polynomial rings $K[x_1, \ldots, x_n]$, and let $\mathcal{R}^{\flat}$ be the subring of $\mathcal{R}$ consisting of elements with bounded denominators. In previous joint work with Erman and Sam, we showed that $\mathcal{R}$ and $\mathcal{R}^{\flat}$ (and many similarly defined rings) are abstractly polynomial rings, and used this to give new proofs of Stillman's conjecture. In this paper, we prove the complementary result that $\mathcal{R}$ is a polynomial algebra over $\mathcal{R}^{\flat}$.