Set-up:
I will use the notation $\underline{S}$ to denote the constant simplicial commutative ring which is $S$ in all degrees and has face and degeneracy maps the identity.
Let $R_{\bullet}$ be a simplicial commutative ring. We may think of $R_{\bullet}$ as a simplicial commutative $\textbf{Z}[x]$-algebra by choosing an element $f\in\pi_{0}R_{\bullet}$ and constructing the map $\underline{\textbf{Z}[x]}\to R_{\bullet}$ which in degree $n$ sends $x$ to the degenerate image of $f$ in $R_{n}$.
Given such an $f\in\pi_{0}R_{\bullet}$, we have the multiplicatively closed subset $W=\{1,f,f^{2},\ldots\}\subseteq\pi_{0}R_{\bullet}$. We can define the localization of $R_{\bullet}$ at $f$ by $$ R_{\bullet}[1/f]:=R_{\bullet}\otimes_{\underline{\textbf{Z}[x]}}^{\textbf{L}}\underline{\textbf{Z}[x,y]/(xy-1)}. $$
Computing a derived tensor product $-\otimes^{\textbf{L}}T_{\bullet}$ means finding a cofibrant replacement $T_{\bullet}^{c}$ of $T_{\bullet}$ and computing $-\otimes T_{\bullet}^{c}$. In any model category, an object $X$ is cofibrant if the unique map $\emptyset\to X$ is a cofibration, where $\emptyset$ is the initial object, and a cofibrant replacement of $T_{\bullet}$ is a cofibrant object $T_{\bullet}^{c}$ that is weakly equivalent to $T_{\bullet}$.
In the same manner that localization of rings is flat, I expect that localization of simplicial commutative rings should satisfy $$ R_{\bullet}\otimes_{\underline{\textbf{Z}[x]}}^{\textbf{L}}\underline{\textbf{Z}[x,y]/(xy-1)}\cong R_{\bullet}\otimes_{\underline{\textbf{Z}[x]}}\underline{\textbf{Z}[x,y]/(xy-1)}; $$ that is, $\underline{\textbf{Z}[x,y](xy-1)}$ is cofibrant, and thus can be taken as its own cofibrant replacement. Also, this should be true for the reason that localization $R_{\bullet}[1/f]$ can also be defined as localization at each degree; i.e., $R_{\bullet}[1/f]:=\left(R_{n}[1/f]\right)_{\bullet}$, writing $f$ for its denegerate image in $R_{n}$. That is what the underived tensor product on the right gives us.
Goerss-Schemmerhorn, in Proposition 4.21 of Model Categories and Simplicial Methods, states that a map in $s\textbf{Alg}_{R}$, the category of simplicial commutative $R$-algebras, is a cofibration if and only if it is a retract of a free morphism. We can define a free morphism in $s\textbf{Alg}_{\textbf{Z}[x]}$ as follows:
Let $\Delta$ be the simplex category with objects $[n]=\{0,\ldots n\}$ and arrows order preserving functions. Let $\Delta_{+}\subseteq\Delta$ be the subcategory with the same objects and arrows only surjective morphisms. Let $R_{\bullet}:\Delta^{op}\to\textbf{Alg}$ be a simplicial algebra. We say $R_{\bullet}$ is $s$-free if the induced object $$ R_{\bullet,+}:\Delta_{+}^{op}\to\textbf{Alg} $$ is free. We say that a map $R_{\bullet}\to S_{\bullet}$ in $s\textbf{Alg}_{\textbf{Z}[x]}$ is $s$-free if $R_{\bullet,+}\to S_{\bullet,+}$ is isomorphic to the inclusion $R_{\bullet,+}\hookrightarrow R_{\bullet,+}\sqcup S'$ where $S'$ is $s$-free. Finally, a map $R_{\bullet}\to S_{\bullet}$ is free if it is $s$-free on a set of objects $\{\mathrm{Sym}^{\bullet}(P_{k})\}$, where each $P_{k}$ is a projective $\textbf{Z}[x]$-module and $\mathrm{Sym}^{\bullet}-$ is the symmetric algebra functor.
Question:
I want to show that $\underline{\textbf{Z}[x,y]/(xy-1)}$ is a cofibrant object in $s\textbf{Alg}_{\textbf{Z}[x]}$.
By definition, that means we must show that $\underline{\textbf{Z}[x]}\to\underline{\textbf{Z}[x,y]/(xy-1)}$ is a cofibration, since $\underline{\textbf{Z}[x]}$ is the initial object in $s\textbf{Alg}_{\textbf{Z}[x]}$.
By Goerss-Schemmerhorn, we must show that $\underline{\textbf{Z}[x]}\to\underline{\textbf{Z}[x,y]/(xy-1)}$ is a retract of a free morphism. That is, we must show that there exists a free morphism $g:A_{\bullet}\to B_{\bullet}$ such that the following diagram commutes, and the horizontal rows compose to give the respective identity maps.
$$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccc} \underline{\textbf{Z}[x]}&\ra{}&A_{\bullet}&\ra{}&\underline{\textbf{Z}[x]}\\ \da{}&&\da{g}&&\da{}\\ \underline{\textbf{Z}[x,y]/(xy-1)}&\ra{}&B_{\bullet}&\ra{}&\underline{\textbf{Z}[x,y]/(xy-1)} \end{array} $$
But I am stuck on how to check this condition. For example, what should $g$ be? Any hints are appreciated, or if there's a better approach!