Animated rings and $\mathbb E_{\infty}$-rings

Question

Let $R$ be a discrete commutative ring. The Proposistion 25.1.2.4 in Lurie's Spectral Algebraic Geometry says that the natural functor from the $\infty$-category of animated $R$-algebras to the $\infty$-category of connective $\mathbb E_{\infty}$-algebras over $R$ admits left and right adjoints. I am wondering if both adjoints are identity on discrete objects?

Answer 1

Let $\DeclareMathOperator\CAlg{CAlg}\DeclareMathOperator\an{an}\CAlg_R^{\an}$ denote the $\infty$-category of animated commutative $R$-algebras, and $\DeclareMathOperator\cn{cn}\CAlg_R^{E_\infty,\cn}$ denote the $\infty$-category of connective $E_\infty$-$R$-algebras.

Proposition 1. The left adjoint to the forgetful functor $\CAlg_R^{\an}\to\CAlg_R^{E_\infty,\cn}$ is identity on usual commutative $R$-algebras if and only if $R$ is a commutative $\mathbb Q$-algebra.

Proof. The "if" part holds since in the char 0 case, the forgetful functor in question is the identity functor. For the "only if" part, note that the left adjoint carries polynomial $R$-algebras to free $E_\infty$-$R$-algebras, Q.E.D.

Proposition 2. The right adjoint to the forgetful functor $\CAlg_R^{\an}\to\CAlg_R^{E_\infty,\cn}$ is identity on usual commutative $R$-algebras.

Proof. The composite functor $$\CAlg_R^{\an}\longrightarrow\CAlg_R^{E_\infty,\cn}\xrightarrow{\tau_{\le0}}\CAlg_R^{\heartsuit}$$ coincides with the truncation $\tau_{\le0}$. Taking right adjoints, we get what we want, Q.E.D.