Reference: the category of derived affine schemes is extensive

Question

The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.

See extensive category. Does this result exist in the literature?

I have difficulty transferring the usual (obvious) proof from discrete rings to simplicial ones, because the product $A \times B$ is not cofibrant. And with cofibrant replacement, idemptotents cease to be strict.

It's surprising if some work is actually needed here, because conceptually the fact should be very simple. I'm probably just missing something.

UPD. I removed the erroneous comment about equivalence with the category of dg algebras, but in reality this result is enough for me for algebras over the field of characteristic 0 (where there is an equivalence with dg algebras), if this, of course, simplifies something.

Answer 1

(First, animated rings are not modeled by connective CDGA's beyond $\mathbb Q$-algebras.)

An $E_\infty$-version is in [Lurie: Spectral Algebraic Geometry, (b) in Proof of Prop B.6.1.6], which implies the animated version. The proof is the same as the classical version: given two commutative ring spectra $A,B$, the unit $1\in\pi_0(A\times B)\cong\pi_0A\times\pi_0B$ is the sum $e_1+e_2$ of two idempotents, where $e_1$ is the image of the unit $1\in\pi_0A$ under the map $\pi_0A\to\pi_0A\times\pi_0B$. The image of $e_1$ (resp. $e_2$) under the map $\pi_0A\times\pi_0B\to\pi_0B$ (resp. $\pi_0A\times\pi_0B\to\pi_0A$) is $0$, thus the image of $1$ under the map $\pi_0A\times\pi_0B\to\pi_0(A\otimes_{A\times B}B)$ is $0$ as well, which shows that the ring $\pi_0(A\otimes_{A\times B}B)$ is $0$, and thus so is the commutative ring spectrum $A\otimes_{A\times B}B$.