Approximating ring maps of finite Tor-dimension

Question

Let $R$ be a commutative ring, and let $S$ be a finitely presented $R$-algebra of finite Tor-dimension over $R$. Can $R \to S$ be realized as the base change, along some ring map $R_0 \to R$, of a finitely presented ring map $R_0 \to S_0$ of finite Tor-dimension with $R_0$ Noetherian? Can we furthermore arrange that $R_0$ is actually of finite type over $\mathbf{Z}$?

Answer 1

As requested in the comments, here is Proposition 6.1.6.1 of Lurie's "Spectral Algebraic Geometry", specialized to the case of ordinary commutative rings:

Let $n\ge 0$ be an integer, let $A_0$ be a commutative ring and let $B_0$ be an $A_0$-algebra of finite presentation. Suppose we are given a diagram $\{A_\alpha\}_{\alpha\in I}$ of $A_0$-algebras indexed by a filtered partially ordered set $I$, and set $A = \varinjlim A_\alpha$. Set $B_\alpha = A_\alpha \otimes_{A_0} B_0$ and $B = A \otimes_{A_0} B_0 \simeq \varinjlim B_\alpha$. If $B$ has Tor-amplitude $\le n$, then there exists an index $\alpha$ such that $B_\alpha$ has Tor-amplitude $\le n$ over $A$.

So the answer to the question is: yes and yes.

Answer 2

First of all, I strongly suspect that the answer to the question is: no.

Secondly, I want to give an example to show that the answer by Riza Hawkeye is incorrect as it stands (see also the comment by Denis Nardin for why the result is quoted incorrectly from Lurie). Namely, consider the system of rings $$ A_0 \to A_1 \to A_2 \to A_3 \to \ldots $$ where $A_i = \mathbf{Z}[x, z_i]/(xz_i)$ and where the maps send $x$ to $x$ and $z_i$ to zero. Then we see that the colimit of the system is $A = \mathbf{Z}[x]$. Now set $B_0 = A_0/xA_0$. Clearly, we see that $B_i = B_0 \otimes_{A_0} A_i$ is equal to $A_i/xA_i$ which has infinite tor dimension over $A_i$ for all $i$. On the other hand, we have $B = A/xA$ and this has tor dimension $1$ as a module over $A = \mathbf{Z}[x]$.