If $A$ is a cofibrant commutative dg-algebra over a commutative ring of characteristic $0$, then its underlying chain complex is cofibrant

Question

Let $R$ be a commutative ring with characteristic $0$, namely it contains the field of rational numbers. Higher Algebra Proposition 7.1.4.10 tells that the category of commutative $R$-dg-algebras $\mathrm{CAlg^{dg}}(R)$ has a model structure induced from the projective model structure on chain complexes $\mathrm{Ch}(R)$, where weak equivalences are quasi-isomorphisms and fibrations are surjections (so, a morphism of commutative dg-algebras is a fibration or a weak equivalence if its underlying morphism of chain complexes is such).

In the proof of the following Proposition 7.1.4.11, an unproved claim (a condition in 4.5.4.7) is implicitly used, namely:

The forgetful functor $\mathrm{CAlg^{dg}}(R) \to \mathrm{Ch}(R)$ preserves fibrant-cofibrant objects.

Now, every object is fibrant with respect to the considered model structures, so this claim boils down to check that if $A$ is a cofibrant object in $\mathrm{CAlg^{dg}}(R)$, then its underlying chain complex is cofibrant with respect to the projective model structure on $\mathrm{Ch}(R)$.

How can I prove this? If $R$ were a field then it would be very easy, because every chain complex over a field is cofibrant. I feel that I should somehow use that $R$ has characteristic $0$ (it contains $\mathbb Q$), but can't precisely figure out how.

Answer 1

Fernando's answer tells you how to prove the statement directly. Alternatively, if you want a reference, this is proven in Corollary 3.6 of my PhD thesis paper (published in JPAA) Model Structures on Commutative Monoids in General Model Categories. I introduce an axiom that a monoidal model category $M$ can satisfy, the "strong commutative monoid axiom," which guarantees that:

1. Commutative monoids in $M$ inherit a model structure transferred from $M$ along the forgetful functor $U$, meaning that a morphism $f$ is a weak equivalence or fibration if and only if $U(f)$ is in $M$, and
2. $U$ preserves cofibrations with cofibrant source.

Note that (1) implies immediately that $U$ preserves fibrant objects. Then, in Section 5.1, I verify that the example you mention does satisfy this axiom. Furthermore, the initial CDGA is cofibrant, so a corollary of (2) is that $U$ takes cofibrant CDGAs to cofibrant chain complexes (sometimes said "$U$ preserves cofibrant objects").

Disclaimer: In Section 5.1 of that paper, I only state the result for when $R$ is a commutative $\mathbb{Q}$-algebra. In a later paper, I make the observation that everything works when $R$ has characteristic zero. Search Homotopical Adjoint Lifting Theorem for "characteristic" to see. This paper was joint with Donald Yau and published in Applied Categorical Structures.

Answer 2

Cofibrant CDGAs are retracts of cellular ones. A cellular cofibrant CDGA is a free commutative graded algebra on a (possibly transfinite) sequence of generators $x_1,x_2,\dots$ such that $d(x_i)$ only depends on previous generators. A linear basis is given by monomials $x_{i_1}^{n_{i_1}}\cdots x_{i_r}^{n_{i_r}}$ such that $i_j<i_{j+1}$ and $n_{i_j}=1$ if $|x_{i_j}|$ is odd. This follows from $\mathbb{Q}\subset R$. You can put a kind of lexicographic order in these monoimials in such a way that the differential of each one only depends on previous monomials. This is cofibrant as a complex in the projective model structure by virtue of the well-known set of generating cofibrations, i.e. it is a cellular complex.