For each scheme or algebraic stack their $\infty$-category of quasicoherent sheaves (resp., perfect complexes) on it is $k$-linear for a commutative ring $k$. That is (by a recent result of L.Cohn), it is a $\mathrm{DG}$-category over $k$.

But what about is the base ring of of $\mathrm{QC(X)}/\mathrm{Perf(X)}$ if $\mathrm{X}$ is a "derived" stack? That is, either

$(1)$ a derived Artin stack in the sense of "derived algebraic geometry" à la Toen and Vezzosi, based on simplicial commutative rings,

$(2)$ or a spectral Deligne-Mumford stack in the sense of "spectral algebraic geometry" à la Lurie, based on $\mathrm{E_{\infty}}$-rings.

There isn't much on the matter in existing papers and literature. The impression I got from browsing some papers on derived (resp., spectral) algebraic geometry, that Toen and Vezzosi consider $\mathrm{QC(X)}/\mathrm{Perf(X)}$ for a derived Artin stack as a $\mathrm{DG}$-category (that is, a $k$-linear $\infty$-category for a commutative ring $k$) while Lurie in his book "Spectral Algebraic Geometry" seems to treat more general $R$-linear $\infty$-categories for $R$ being an $\mathrm{E_{\infty}}$-ring (however, $\infty$-categories arising from spectral Deligne-Mumford stacks are still stable).

What I'm trying to ask is the following: Let $\mathrm{X}$ be a derived Artin stack or a spectral Deligne-Mumford stack. Then $\mathrm{QC(X)}$ and $\mathrm{Perf(X)}$ are $R$-linear (stable) $\infty$-category. What is $R$? An ordinary ring? A simplicial commutative ring? An $\mathrm{E_{\infty}}$-ring? Or what is the "base ring" of a derived stack?

Also, perhaps the better question would be the following:
**Given a derived (resp., spectral) stack $\mathrm{X}$, what $R$ can be "over"?**

For example, derived Artin stacks are "built" from simplicial $k$-algebras for $k$ being a commutative ring. So even if there is a leap from commutative rings to simplicial commutative rings, ordinary commutative rings are still involved.

**P.S.** I assume that this question may seem a little naive for people who are familiar with derived algebraic geometry, but I don't know a lot on the matter, so forgive me if the question is of the low quality.