Cofibrant CDGAs are retracts of standard ones. A standard 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.

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.

Cofibrant CDGAs are retracts of standard ones. A standard 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. 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.

Cofibrant CDGAs are retracts of standard ones. A standard 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.