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I want to figure out the following question: How to get an $E_\infty$-ring from a commutative differential graded ring?

More precisely, let $\operatorname{cdga}$ be the ($1$-)category of cdgas, let $E_\infty\text{-ring}$ denote the $\infty$-category of $E_\infty$-rings. I'm searching for a functor $f\colon\operatorname{cdga}\rightarrow E_\infty\text{-ring}$ that satisfies the following properties:

(i) The composition functor $\operatorname{cdga}\stackrel{f}{\rightarrow} E_\infty\text{-ring}\rightarrow\operatorname{Sp}$ should be equivalent to the composition functor $\operatorname{cdga}\rightarrow\operatorname{Ch}(\mathbb{Z})\rightarrow\operatorname{Ch}(\mathbb{Z})[w^{-1}]\cong\mathcal{D}(\mathbb{Z})\rightarrow\operatorname{Sp}$.

(ii) The functor $f$ should give rise to the correct graded ring structure on the homotopy groups.

I've searched in the literature but failed to find an explicit reference. HA.7.1.4.6 seems to solve the associative case, but the analogous statement for the commutative case (HA.7.1.4.11) requires us to work over $\mathbb{Q}$.

I think a possible way to proceed is to prove that the functor $\operatorname{Ch}(\mathbb{Z})\rightarrow\operatorname{Ch}(\mathbb{Z})[w^{-1}]$ is lax symmetric monoidal (where the symmetric monoidal structure on the latter $\infty$-category is provided by HA.4.1.7.6), but I don't know how to prove this statement.

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  • $\begingroup$ There is no functor satisfying these requirements. $\endgroup$
    – Ben Wieland
    Commented Jun 5 at 14:50
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    $\begingroup$ The functor $\mathrm{Ch}(\mathbb Z)\to \mathrm{Ch}(\mathbb Z)[w^{-1}]$ is in fact lax symmetric monoidal. This is proved in my paper "On topological cyclic homology" with Thomas Nikolaus, see Theorem A.7. $\endgroup$
    – Peter Scholze
    Commented Jun 5 at 15:36

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There is such a functor constructed in [Arpon Raksit: Hochschild homology and the derived de Rham complex revisited, Example 3.3.14]. It works as follows:

Let $k$ be a (usual) commutative ring. Then the symmetric monoidal $\infty$-category of completely filtered $k$-module spectra has a compatible $t$-structure, called the Beilinson $t$-structure, whose heart coincides with the $1$-category of cochain complexes. The compatibility implies that the inclusion of the heart has a lax symmetric monoidal structure, thus a CDGA gives rise to a filtered $E_\infty$-$k$-algebra. Passing to the underlying $E_\infty$-algebra, you get what you want.

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This is page 25 of Lurie's Derived algebraic geometry. I'm not sure whether this is what you need. enter image description here

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    $\begingroup$ Lurie doesn't seem to provide an explicit construction of the functor $\psi$... $\endgroup$
    – Yebo Peng
    Commented Jun 5 at 14:07

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